Scalable variational learning for noisy-OR Bayesian networks with normalizing flows for complex cascading disaster systems

Abstract

Sudden-onset disasters like earthquakes trigger multiple cascading hazards and impacts, causing human and economic losses. While remote sensing technologies enable rapid hazard assessment, current methods have two key limitations: they struggle to decouple co-located hazards and impacts, and cannot adapt to new information during disaster response. We present online-DisasterVINF, a framework addressing both challenges through Noisy-OR Bayesian networks and normalizing flows. This approach models how multiple hazards independently contribute to observed impacts while providing a more expressive alternative to log-linear relationships. The framework continuously incorporates ground truth data from post-disaster reconnaissance through a novel variational inference approach that jointly approximates posteriors by leveraging causal relationships and remote sensing techniques. Evaluation across multiple global seismic events demonstrates that online-DisasterVINF significantly improves estimation accuracy compared to existing methods, while showing substantial performance enhancement as ground truth becomes available, highlighting its utility for real-time disaster response.

Introduction

Moderate-to-large-scale disasters, particularly earthquakes, often trigger a cascade of secondary hazards such as landslides and liquefaction. These, combined with ground shaking, lead to significant impacts including building damage and loss of life. For example, the 2023 Turkey-Syria earthquake with magnitude 7.8 caused more than 55,000 deaths, displaced 3 million people in Turkey and 2.9 million in Syria, and destroyed or damaged at least 230,000 buildings. Additionally, post-disaster reporting identified thousands of landslides (with initial satellite imagery showing over 100 landslides) and 760 sites with liquefaction and lateral spreading features^1,2,3. Rapid, accurate assessment of the location and severity of these disaster-induced hazards and impacts is crucial for effective victim rescue within the critical “Golden 72 Hour" period⁴ and for guiding post-disaster reconnaissance and recovery efforts.

With the surging interest in rapid disaster responses, researchers have developed various approaches to estimate post-disaster hazards and losses^{5,6,7,8,9,10,11}. Current practices primarily focus on estimating single-type hazards, utilizing statistical models calibrated against historical inventories with geospatial susceptibility proxies (e.g., slope and lithology) and ground motion intensity measures. For example, building fragility curves and vibration-based machine learning models help model the seismic vulnerability of buildings under different seismic intensities^12,13,14. However, these approaches suffer from low accuracy and high uncertainty due to overlapping events, simplified modeling, limited building type information, and substantial uncertainties of geological conditions.

Recent advancements in remote sensing techniques, particularly Interferometric Synthetic Aperture Radar (InSAR), have enabled high-resolution, on-site information for rapid hazard estimation^15,16. The Advanced Rapid Imaging and Analysis (ARIA) system processes SAR data into Damage Proxy Maps (DPMs), which can infer ground-surface changes from satellite distance changes^15,17,18. However, when a disaster triggers multiple, simultaneous hazards, these causally linked processes result in complex, mixed signals in InSAR images, making interpretation challenging. Consequently, supervised models using InSAR imagery signals often struggle with accuracy and generalization^15,18,19.

Recent research has highlighted the importance of considering these complex interactions through multi-hazard risk assessment frameworks^20,21,22, with Bayesian networks emerging as a promising direction^{23,24,25,26,27,28,29,30,31,32,33}. However, existing approaches face two critical limitations: they either lack explicit causal structures (e.g., ref. ³⁴) or rely on simple log-linear relationships that cannot capture complex interdependencies. Moreover, these models remain static even as valuable ground truth data becomes available during post-disaster reconnaissance, missing opportunities to improve their estimation with real-world observations. Therefore, there remains a pressing need for physically interpretable modeling approaches that can examine cascading hazards holistically and disentangle complex signals in remote sensing data.

The limitations of existing approaches and the complexity of disaster scenarios highlight several fundamental challenges in developing effective disaster response systems. Key challenges posed by this goal include (1) integrating heterogeneous ontologies—including prior physical knowledge, multiple hazards, impacts, environmental noise, and remote sensing observations—into a holistic model that best approximates the true geo-process is complex. (2) The highly intricate causal dependencies among cascading hazards and impacts require expressive modeling approaches that can capture their interactions. (3) Developing models that can not only begin inference without ground truth data but also adaptively improve as such data becomes incrementally available through post-disaster reconnaissance is particularly challenging. (4) Achieving near real-time responses requires efficient computation of high-resolution hazard and impact distributions across large-scale disaster zones.

In this paper, we introduce online-DisasterVINF, an accurate, physically interpretable, and near real-time framework for jointly estimating cascading hazards and impacts from satellite imagery. Our framework integrates prior physics-informed models, cascading hazards, impacts, environmental noises, and remote sensing observations into a causal graph, connected by their physical dependencies, to best approximate the disaster chain. We formulate this causal graph as a novel family of causal Bayesian networks (BNs) with a noisy-OR structure, which is particularly suited for modeling complex disaster chains where multiple causes independently influence outcomes. The noisy-OR condition, which is common in physical models, assumes each child variable is independently influenced by each parent, enabling efficient capture of higher-order relations and complex interdependencies among multiple hazards and impacts while maintaining the interpretability of the causal mechanisms^35,36. To better approximate the complex process, we model the causal dependencies in the BN using normalizing flows^28,37. Much like how a natural disaster gradually transforms the landscape through a series of changes, normalizing flows transform simple probability distributions into more complex ones through a sequence of mathematical operations. Each transformation in the flow builds upon previous ones while preserving important relationships, allowing us to capture intricate patterns in disaster processes that simpler models might miss. This approach provides a parameter-efficient and expressive tool for constructing complex transformations through a series of invertible mappings. We design a triplet loss to constrain the divergence between prior physical knowledge and posteriors, improving the identifiability of our highly expressive BNs, and develop a stochastic variational inference algorithm to jointly infer posteriors of unobserved variables in the BNs efficiently and accurately without ground truth data, by maximizing variational lower bounds of the marginal likelihood of observed remote sensing signals. To enable real-time processing of large-scale disasters, we implement a local pruning strategy to effectively reduce complexity in causal graph processing, making online-DisasterVINF particularly suitable for processing vast amounts of data across extensive geographical areas. Furthermore, we incorporate an online updating mechanism that adaptively improves estimation as ground truth data becomes incrementally available post-disaster, bridging the gap between initial rapid assessments and refined analyses that incorporate field observations.

Methods

Our framework addresses the challenge of jointly modeling and estimating cascading hazards and impacts through three key technical innovations: (1) a novel causal Bayesian network structure with noisy-OR gates to model-independent hazard influences, (2) integration of normalizing flows to capture complex causal dependencies, and (3) an online learning mechanism that enables real-time refinement as ground truth data becomes available.

Evaluation metrics

We evaluate our framework using the receiver operating characteristics (ROC) curve and its area under curve (AUC) metric³⁸. The ROC curve, constructed by plotting true positive rate (TPR) against false positive rate (FPR) across varying decision thresholds, provides a threshold-independent assessment critical for disaster scenarios where optimal classification thresholds may vary by event. This evaluation approach is particularly suitable for our framework and benchmarks, which output probabilistic estimates rather than binary classifications. While our online-DisasterVINF is fundamentally an unsupervised model that approximates true posteriors through variational inference, the ROC analysis enables objective comparison with supervised approaches. Additionally, we utilize the variational bound to determine the optimal flow length.

Causal Bayesian network framework

To develop a comprehensive approach for jointly estimating multi-hazards and impacts with adaptation capability, we begin by constructing a generalized causal graph, which we then formulate into a causal Bayesian Network (BN). This process involves identifying key factors as nodes and their qualitative physical dependencies as edges, thereby creating a representation that best captures the cascading nature of disaster processes. Figure 1 illustrates this concept in the context of seismic events. For instance, an initial earthquake triggers ground shaking, which in turn can lead to secondary hazards such as landslides and liquefaction. These primary and secondary hazards collectively contribute to impacts like building damage. The entire chain of events, from the initial trigger to the final impacts, can be captured through various sensing observations, including satellite imagery and Damage Proxy Maps (DPMs).

Fig. 1: An example cascading causal chain of how an earthquake induces multiple hazards and impacts is further jointly captured by sensing observations.

The causal chain can also be expanded to other disasters like hurricanes and tsunamis.

Unlike previous approaches, our network structure is designed to be dynamically updatable, preparing it for the integration of post-disaster ground truth data. Figure 2 depicts the resulting BN, which integrates random variables representing prior information, unobserved hazards and impacts, and sensing observations. Ground shaking, representing the peak ground acceleration (PGA)³⁹, plays a fundamental role in our network by influencing hazard occurrence through multiple pathways. It directly informs the prior geospatial models (α_LS and α_LF) alongside other geospatial features, while also having a direct causal relationship with building damage through ${r}_{\rm{BD}}^{0}$. This dual-path structure captures both the physics-based relationships between ground motion and hazard susceptibility, as well as the independent effects of seismic intensity on building damage.

Fig. 2: An example causal Bayesian network for multi-hazards and impacts estimation in seismic events.

N in the figure refers to N locations in a target area. Green rectangles refer to the known variables. Blue circles refer to unobserved or unknown nodes. γ_i are the unknown causal parameters that quantify the causal relations among parent and child nodes. η_j are the unknown parameters in normalizing flows.

In the context of seismic events, we focus on three critical types of seismic damage: landslides (LS), liquefaction (LF), and building damage (BD). The sensing observations in our model primarily consist of satellite observations and DPMs. Our Bayesian network architecture incorporates two distinct types of vertices: (1) Feature vertices that represent the random variables in the causal graph, encompassing hazards, impacts, and observations; and (2) weight vertices that parameterize the causal links between nodes, quantifying the strength and nature of the relationships in the network. This structure allows us to model both the presence of relationships between different elements of the disaster process and the strength and characteristics of these relationships. By doing so, we create a flexible and comprehensive framework for estimating the complex interactions that occur during and after a seismic event, while maintaining the capability to refine these relationships as new information becomes available.

Node variables modeling

We consider a target area comprising N locations, each indexed by l. Typically, a location corresponds to a pixel in remote sensing observations. For each location l, we define variables of unobserved hazards and impacts as z ∈ Z (omitting the index l for simplicity). We denote z_i as the i^th unobserved variable in Z, where i ∈ {LS, LF, BD} as illustrated in Fig. 2. It is important to note that these unobserved variables z_i are binary, indicating the presence or absence of specific hazards or impacts.

We use ${\mathcal{P}}(i)$ to represent the set of parent nodes for z_i. To account for environmental factors, we introduce ϵ_i to represent environmental noise for node z_i. Prior geospatial model estimates are denoted as α, while sensing observations are represented by x ∈ X. Specifically, x^l refers to the damage proxy map (DPM) pixel captured at location l, and ${\mathcal{P}}(x)$ denotes the parents of sensing observations X. To enhance the flexibility of our framework, we introduce a leak node, z_0i = 1, which is always active. This allows for a non-zero probability of a child node being activated even when all other parent nodes are inactive.

Causal dependency modeling with normalizing flow

One of the most challenging aspects of our work is constructing expressive quantitative causal dependencies that can accurately approximate complex physical processes while adapting to new information. This challenge is represented by the edges shown in Fig. 7. To understand this challenge, consider how an earthquake triggers various hazards: the relationship between ground shaking and landslide probability is not simply linear—it depends on multiple factors like soil conditions, slope stability, and moisture content, all interacting in complex ways. Traditional approaches often oversimplify these relationships using linear models, which limits their ability to capture real-world complexity.

Normalizing flows provides a powerful solution to this challenge by transforming simple probability distributions into more complex ones through a sequence of invertible operations. This process is analogous to the process of gradually transforming a basic shape into an intricate form: each small adjustment builds upon previous changes while maintaining the essential structure. In our context, we start with a simple Gaussian distribution and apply a series of carefully designed transformations to create more expressive distributions that better represent the complex relationships between hazards and their impacts.

These transformations work similarly to how different hazards combine to influence observed damage. For instance, both landslides and ground shaking can damage buildings, but their combined effect is not simply additive - they interact in complex ways that change with local conditions. Our flow-based approach captures these intricate relationships while maintaining the ability to update as new information becomes available during post-disaster reconnaissance.

We denote w_ji $(j\in {\mathcal{P}}(i))$ as the weight of the impacts of each parent node on the child node. Traditional methods often define logit-linear relationships $\log \frac{p({z}_{i}=1| {\mathcal{P}}(i),{\epsilon }_{i})}{1-p({z}_{i}=1| {\mathcal{P}}(i),{\epsilon }_{i})}={\sum }_{j\in {\mathcal{P}}(i)}{w}_{ji}{z}_{j}+{w}_{{\epsilon }_{i}}{\epsilon }_{i}+{w}_{0i}$.

To overcome these limitations, we introduce normalizing flows with adaptive capabilities to model these complex causal relationships. We first define an auxiliary variable ${r}_{i}^{0}$ for each target unobserved variable z_i following a Gaussian distribution: ${r}_{i}^{0}| {\mathcal{P}}(i) \sim {\mathcal{N}}({\mu }_{i}^{0}| {\mathcal{P}}(i),{({\sigma }_{i}^{0})}^{2}| {\mathcal{P}}(i))$. Through normalizing flows, we construct complex, adaptable distributions that can better represent the evolving nature of disaster processes. The flow transformations are designed to not only capture complex physical relationships but also accommodate updates as new information becomes available. For the initial distribution parameters, we define ${\mu }_{i}^{0}=\phi ({{\bf{Z}}}_{{\mathcal{P}}(i)},{\epsilon }_{i};{{\bf{W}}}_{i,\mu }),{({\sigma }_{i}^{0})}^{2}=\delta ({{\bf{Z}}}_{{\mathcal{P}}(i)},{\epsilon }_{i};{{\bf{W}}}_{i,\sigma })$, where ϕ( ⋅ ) represents an adaptive linear combination of parent nodes weighted by W_i,μ, and δ( ⋅ ) combines parent nodes weighted by W_i,σ before applying a Softplus activation to ensure a positive variance. Unlike static approaches, these mapping functions can be updated to reflect a new understanding of hazard relationships.

When applying transformations to variable r with distribution q(r), the resulting variable ${r}^{\prime} ={f}(r)$ follows $q({\bf{r}}^{\prime} )=q({\bf{r}})| \,\text{det}\,\frac{\partial {f}^{-1}}{\partial {\bf{r}}{\prime} }| =q({\bf{r}})| \,\text{det}\,\frac{\partial f}{\partial {\bf{r}}{\prime} }{| }^{-1}$. Through K successive transformations, we construct a variable with complex, refineable distributions as ${r}_{i}^{K}={f}_{K}\,{\circ}\, \cdots\; {\circ}\; {f}_{1}({r}_{i}^{0})$.

The final distribution q_i of our target unobserved variable z_i follows $\log \frac{p({z}_{i}=1| {\mathcal{P}}(i),{\epsilon }_{i})}{1-p({z}_{i}=1| {\mathcal{P}}(i),{\epsilon }_{i})}={r}_{i}^{K}$. To obtain posterior probabilities that can be refined over time, we employ Monte Carlo sampling. For each node i, we sample ${r}_{i}^{K} \sim q({r}_{i}^{K})$ with size M to compute:

$${q}_{i}=\frac{1}{M}\sum\limits_{j = 1}^{M}\frac{1}{1+\exp (-{r}_{ij}^{K})}$$

(1)

This flow-based approach provides several advantages over static models: (1) it captures complex physical dependencies through expressive transformations, (2) its parameters can be updated as new information becomes available, and (3) it maintains the interpretability of the physical relationships even as they evolve.

For sensing observations, we model the relationship between hazards/impacts and observations using an adaptive log-normal distribution as $x| {\mathcal{P}}(x),{\epsilon }_{x} \sim LN({\sum }_{i\in {\mathcal{P}}(x)}{w}_{ix}{z}_{i}+{w}_{{\epsilon }_{x}}{\epsilon }_{x}+{w}_{0x},{w}_{{\epsilon }_{x}}^{2})$. The causal links among nodes are quantified through a set of updatable parameters, represented by our weight vertices. For notational clarity, w_ji $(\,j\in {\mathcal{P}}(i))$ represents the causal relation from parent node j to child node i, including noise terms. These weights (w_ji = γ, η in Fig. 2 for LS, LF, BD, and DPM) remain consistent across locations l ∈ N but can be refined as ground truth data becomes available.

This enhanced formulation differs from previous approaches in several key aspects. The flow transformations are designed to be updatable rather than static, while the parameter space accommodates the temporal evolution of relationships. Additionally, the sampling process can incorporate new information as it becomes available, and the observation model maintains physical interpretability while allowing refinement. This combination of adaptability and interpretability makes our framework particularly suitable for real-world disaster scenarios where understanding evolves as post-disaster reconnaissance data becomes available.

Inference algorithm with online adaptation

Our methodology advances beyond traditional inference approaches through three key innovations: (1) a stochastic variational inference framework with normalizing flows that can adapt to new information, (2) an enhanced triplet loss mechanism that balances prior physical knowledge with emerging ground truth data, and (3) an efficient computation strategy combining local pruning with online updating capabilities.

Adaptive variational inference framework

We develop a flexible inference framework that can evolve as post-disaster information becomes available. The framework transforms the inference problem into an optimization task while maintaining the ability to incorporate new ground truth data. We derive a lower bound of the marginal log-likelihood of observed data that can be continuously refined:

$$\log p(X) \geq {\mathcal{L}}_{v} = \underbrace{{\mathbb{E}}_{q({\textbf{Z}},\epsilon)}[\log p(x,{\mathbf{Z}}, \epsilon)]}_{[1]} - \underbrace{{\mathbb{E}}_{q({\textbf{Z}},\epsilon)}[\log q({\mathbf{Z}},\epsilon)]}_{[2]}$$

(2)

The posterior model q obtained from our flow-based model is designed to be updatable, allowing for refinement as new information becomes available. To derive the complete form of this adaptive bound, we expand item [1]:

$${\mathbb{E}}[\log p(x,{\mathbf{Z}}, \epsilon)] =\underbrace{{\mathbb{E}}(\log p(x|{\mathbf{Z}}_{{\mathcal{P}}(x)},\epsilon_{x}))}_{[3]}+\underbrace{\mathop{\sum}\nolimits_{i}{\mathbb{E}}[\log p\left(z_{i}|\epsilon_{i},{\mathcal{P}}(i)\right.]}_{[4]} + C_{1}$$

For item [2] in Eq. (2), we derive the expectation with respect to our adaptive flow-based model. The transformed density q_i follows an evolving distribution that can incorporate new information:

$$\begin{array}{ll}&{r}_{i}^{K}={f}_{K}\,{\circ}\;{f}_{K-1}{\circ}\;{\cdots}\;{\circ}\,{f}_{1}({r}_{i}^{0})\\ &\log q({r}_{i}^{K})=\log q({r}_{i}^{0})-\sum\limits_{k = 1}^{K}\log\left|1+{{\bf{u}}}_{k}^{T}{\psi }_{k}({{\bf{r}}}_{i}^{k})\right|\end{array}$$

(3)

This allows us to compute expectations with respect to the transformed density q_K through:

$${{\mathbb{E}}}_{{q}_{K}}[h(z)]={{\mathbb{E}}}_{{q}_{0}}[h(\,{f}_{K}\,{\circ}\, \cdots {\circ}\;{f}_{1}({{\bf{z}}}_{0}))]$$

(4)

where z₀ follows a base distribution q₀ that can be updated with new observations.

The reparameterization of ${r}_{i}^{0}$ follows: ${r}_{i} \sim {\mathcal{N}}({r}_{i}| \mu ,{\sigma }^{2})\iff {r}_{i}=\mu +\sigma \epsilon ,\epsilon \sim {\mathcal{N}}(0,1)$, where {μ, σ²} are derived through transformations of both prior physical models and available ground truth.

The expectation in item [2] becomes:

$$\begin{array}{ll}&{{\mathbb{E}}}_{q({\bf{Z}},\epsilon )}[\log q({\bf{Z}},\epsilon )]=\sum\limits_{i\in \{BD,LS,LF\}}\left[{\mathbb{E}}(\log q({r}_{i}^{0}))-{\mathbb{E}}\left(\sum\limits_{k = 1}^{K}\log | \det\frac{\mathop{\partial{f}}\nolimits_{d}^{k}}{\mathop{{\partial}{r}}\nolimits_{i}^{k-1}}|\right)\right]\end{array}$$

(5)

For sensing observations, we expand item [3]:

$$\begin{array}{ll}{\mathbb{E}}(\log p(x| {{\bf{Z}}}_{{\mathcal{P}}(x)},{\epsilon }_{x}))=-\log x-\log | {w}_{{\epsilon }_{x}}|\\\qquad\qquad\qquad\qquad\quad-\,\displaystyle\frac{\sum\nolimits_{j\ne l}{w}_{jx}{w}_{lx}{\mathbb{E}}({q}_{j}){\mathbb{E}}({q}_{l})}{{w}_{{\epsilon }_{x}}^{2}}-\frac{{(\log x)}^{2}+{w}_{0x}^{2}+\sum\nolimits_{j}{w}_{jx}^{2}{\mathbb{E}}({q}_{j})}{2{w}_{{\epsilon }_{x}}^{2}}\\\qquad\qquad\qquad\qquad\quad-\,\displaystyle\frac{({w}_{0x}-\log x)\sum\nolimits_{j}{w}_{jx}{\mathbb{E}}({q}_{j})-{w}_{0x}\log x}{{w}_{{\epsilon }_{x}}^{2}}\end{array}$$

(6)

For unobserved variables, item [4] follows our adaptive noisy-OR formulation:

$$\begin{array}{l}{\mathbb{E}}[\log p\left({z}_{i}| {\epsilon }_{i},{\mathcal{P}}(i)\right.]\,=\,{\mathbb{E}}({q}_{i}){\mathbb{E}}[-\log (1+\exp (-su{m}_{i}))]\\\qquad\qquad\qquad\qquad\;\,+\,(1-{\mathbb{E}}({q}_{i})){\mathbb{E}}[-\log (1+\exp (su{m}_{i}))]\end{array}$$

(7)

where $su{m}_{i}=\sum _{j\in {\mathcal{P}}(i)}{w}_{ji}{z}_{j}+{w}_{{\epsilon }_{i}}{\epsilon }_{i}+{w}_{0i}$.

For the intractable log-sum-exp term, we derive a dynamic lower bound:

$${\mathbb{E}}[-\log (1+\exp (x))]\,\ge -\log (1+{\mathbb{E}}(\exp (x)))$$

(8)

This leads to refined bounds for hazard variables (i ∈ {LS, LF}):

$$\begin{array}{ll}&{\mathbb{E}}[\log p\left({z}_{i}| {\epsilon }_{i},{\mathcal{P}}(i)\right.]\\ &\ge {\mathbb{E}}({q}_{i})\left[-\log\left(1+\exp\left(\frac{{w}_{{\epsilon }_{i}}^{2}}{2}-{w}_{0i}-{w}_{{\alpha }_{i}}{\alpha }_{i}\right)\right)\right]\\ &\;+\,(1-{\mathbb{E}}({q}_{i}))\left[-\log\left(1+\exp\left(\frac{{w}_{{\epsilon }_{d}}^{2}}{2}+{w}_{0i}+{w}_{{\alpha }_{i}}{\alpha }_{i}\right)\right)\right]\end{array}$$

(9)

Physical constraints and model identifiability

Our framework incorporates physical constraints through XOR nodes that can be updated based on new observations. For mutually exclusive hazards like landslides and liquefaction, we define:

$$p(g=1)={q}_{a}(1-{q}_{b})+{q}_{a}(1-{q}_{b})$$

(10)

Due to the exclusive assumption between landslides (z_LS) and liquefaction (z_LF) and the distribution of the XOR node in Eq. (10), we can further derive the lower bound in Eq. (7) for d = BD, which is shown in Eq. (11):

$$\begin{array}{ll}&{\mathbb{E}}[\log p({z}_{\rm{BD}}| {\epsilon }_{\rm{BD}},{\mathcal{P}}({\rm{BD}}))]\\ &\ge {\mathbb{E}}({q}_{\rm{BD}})\left[-\log\left(1+\exp\left(\displaystyle\frac{{w}_{\rm{BD}}^{2}}{2}-{w}_{0{\rm{BD}}}-{w}_{{\alpha }_{\rm{BD}}}{\alpha }_{\rm{BD}}\right)\right)\right.\\ &\left.\cdot (\exp (-{w}_{g})p(g=1)+p(g=0))\right]\\ &+(1-{\mathbb{E}}({q}_{\rm{BD}}))\left[-\log\left(1+\exp\left(\displaystyle\frac{{w}_{\rm{BD}}^{2}}{2}+{w}_{0{\rm{BD}}}+{w}_{{\alpha }_{\rm{BD}}}{\alpha }_{\rm{BD}}\right)\right)\right.\\ &\left.\cdot (\exp ({w}_{g})p(g=1)+p(g=0))\right]\end{array}$$

(11)

To enhance model identifiability while allowing for adaptation, we employ a dynamic triplet loss:

$${{\mathcal{L}}}_{p}(a,b)=\max \{0,\parallel {q}_{a}-{\rm{ff}}_{a}\parallel -\parallel {q}_{b}-{\rm{ff}}_{a}\parallel +{\rm{fi}} \}$$

(12)

Therefore, the complete variational bound has the following form:

$$\begin{array}{lll}{\mathcal{L}}_{v}&=&-{\rm{log}}\,x-{\rm{log}}|w_{\epsilon_{x}}| - \displaystyle{\frac{\mathop{\sum}\nolimits_{j \ne l \in \{{\rm{LS}},{\rm{LF}},{\rm{BD}}\}}w_{jx}w_{lx}{\mathbb{E}}(q_{j}){\mathbb{E}}(q_{l})}{w_{\epsilon_{x}}^{2}}}\\&&-\,\displaystyle\frac{{({\rm{log}}\,x)}^{2} + w_{0x}^{2} + \mathop{\sum}\nolimits_{i\in\{{\rm{LS}},{\rm{LF}},{\rm{BD}}\}}w_{ix}^{2}{\mathbb{E}}(q_{i})}{2w_{\epsilon_{x}}^{2}}\\&&-\,\displaystyle\frac{(w_{0x}-{\rm{log}}\,x)\mathop{\sum}\nolimits_{i\in\{{\rm{LS}},{\rm{LF}},{\rm{BD}}\}}w_{ix}{\mathbb{E}}(q_{i}) - w_{0x}{\rm{log}}\,x}{w_{\epsilon_{x}}^{2}}\\&&-\,\mathop{\sum}\limits_{i\in \{{\rm{BD}},{\rm{LS}},{\rm{LF}}\}}\left[{\mathbb{E}}({\rm{log}}\,q(r_{i}^{0}))-{\mathbb{E}}\left(\mathop{\sum}\limits_{k = 1}^{K}\log |{\text{det}}\frac{\partial f_{i}^{k}}{\partial z_{i}^{k-1}}|\right)\right]\\&&+\,\mathop{\sum}\limits_{i \in \{\rm{LS,LF}\}}\left\{{\mathbb{E}}(q_{i})\left[-{\rm{log}}\left(1+{\rm{exp}}\left(\frac{w_{\epsilon_{i}^{2}}}{2} - w_{0i}-w_{\alpha_{i}}\alpha_{i}\right)\right)\right]\right.\\&&\left.+(1-{\mathbb{E}}(q_{i}))\left[-\log\left(1+{\rm{exp}}\left(\frac{w_{\epsilon_{i}^{2}}}{2} + w_{0i} + w_{\alpha_{i}}\alpha_{i}\right)\right)\right] \right\}\\&&+\,{\mathbb{E}}(q_{{BD}})[-{\rm{log}}\left(1+{\rm{exp}}\left(\frac{w_{{BD}}^{2}}{2} - w_{0{{BD}}} - w_{\alpha_{{BD}}}\alpha_{{BD}}\right)\right)\cdot ({\rm{exp}} (-{w}_{g}){p}({g}=1)+{p}({g}=0))]\\&&+\,(1-{\mathbb{E}}(q_{{BD}}))[-{\rm{log}}\left(1+{\rm{exp}}\left(\frac{w_{{BD}}^{2}}{2} +{w}_{{0}{{BD}}}+{w}_{\alpha_{{BD}}}\alpha_{{BD}}\right)\right)\cdot({{\rm{exp}}}({w}_{g}){p}({g} = 1)+{p}({g}=0))]\\&&+\,{{\rm{max}}}\{0, \Vert q_{\rm{LS}} - \alpha_{\rm{LS}}\Vert - \Vert q_{\rm{LF}} - \alpha_{\rm{LS}} + \beta\Vert\} + {{\max}}\{0, \Vert q_{\rm{LF}} - \alpha_{\rm{LF}}\Vert -\Vert q_{\rm{LS}} - \alpha_{\rm{LF}} + \beta\Vert\}\end{array}$$

(13)

where q_i is define in Eq. (1).

Adaptive optimization strategy

Our optimization approach employs a modified expectation–maximization algorithm that can incorporate new information during the inference process. In each iteration, the algorithm samples locations considering both regions with and without ground truth data. It then updates posterior estimates using current parameters and available ground truth information, followed by optimizing parameters using the combined information from all sources. The weight updates follow:

$$\begin{array}{l}{{\bf{w}}}_{(t+1)}\,=\,{{\bf{w}}}_{(t)}+\rho A\nabla {\mathcal{L}}({\bf{w}}),\\\;{{\boldsymbol{\tau }}}_{(t+1)}\,=\,{{\boldsymbol{\tau }}}_{(t)}+\rho A\nabla {\mathcal{L}}({\boldsymbol{\tau }}),\end{array}$$

(14)

Efficient computation with dynamic pruning

We implement an adaptive pruning strategy that maintains efficiency while accommodating new information. The algorithm dynamically identifies active nodes based on current data and ground truth, while selecting relevant parent nodes for the inference process. This dynamic approach continuously updates the inference network structure as new observations become available, ensuring the framework can adapt to changing conditions. Our implementation leverages Google Colab’s cloud-based environment, equipped with an NVIDIA Tesla T4 GPU (15 GB VRAM) and 51 GB of RAM. This cloud-based infrastructure proves sufficient for processing large-scale disaster zones within operationally relevant timeframes—analyzing the Haiti earthquake region (237 × 67 km²) in 8821 s and the Puerto Rico region (48 × 27 km²) in 713 s. These processing times align well with typical satellite revisit intervals, which range from hours to 2 days. The efficiency of our framework is enhanced through our local pruning strategy. Through this pruning mechanism and cloud-based implementation, we maintain computational efficiency while preserving the ability to incorporate new ground truth data, enabling effective processing of large-scale disaster zones with evolving information.

Online updating mechanism with partial ground truth

Real-world disaster response scenarios present a unique challenge: ground truth data becomes available gradually through post-disaster reconnaissance efforts, yet rapid assessment needs persist throughout the response period. We develop online-DisasterVINF, an extension of our framework that seamlessly integrates new ground truth data while maintaining continuous operational capability.

Using the 2021 Haiti earthquake as our case study, we evaluate the effectiveness of this mechanism with building damage data collected between April 5–26, 2022. The temporal metadata in this dataset provides a natural experiment for studying how our model adapts to incrementally available information - a critical feature for real-world disaster response scenarios where understanding evolves over time.

To rigorously validate our online learning mechanism, we design an evaluation protocol that mirrors real-world data collection patterns. For handling limited ground truth data, the building damage data is first chronologically sorted based on collection timestamps, then formed into cumulative subsets representing increasing percentages of the total ground truth. This progressive structure reflects the reality of post-disaster reconnaissance, where information accumulates gradually through field surveys and damage assessments. For evaluating robustness to noisy data, we utilize our synthetic data generation process to create controlled test scenarios with varying levels of noise in the ground truth labels.

Our framework implements a novel hybrid inference strategy that seamlessly combines initial estimates with emerging ground truth data. The strategy accounts for both data quantity and quality challenges through the modified variational inference algorithm that adapts its behavior based on data availability at each location. For areas where ground truth becomes available, the model directly incorporates these observations while simultaneously leveraging this new information to refine estimates in regions still lacking ground truth data. When ground truth data contains potential noise, the algorithm balances these observations against estimation from our physics-informed prior models. This approach enables continuous refinement without disrupting ongoing assessment capabilities.

The technical implementation centers on an adaptive expectation–maximization (EM) algorithm. The expectation step dynamically adjusts posterior estimates based on data availability: directly using ground truth values where available while maintaining normalizing flow-based inference elsewhere. The design of our algorithm inherently handles noisy ground truth through its weighted combination approach—the maximization step computes gradients using a weighted combination of ground truth observations and inferred values, allowing the model to learn from both sources simultaneously while maintaining robustness to noise. This unified approach creates a framework that operates continuously without distinct training phases, seamlessly integrates new information while preserving initial rapid assessment capability, and progressively refines its causal understanding. Through transfer learning effects, the model can even improve accuracy in areas lacking ground truth data by leveraging patterns learned from observed locations.

The resulting framework bridges the gap between immediate post-disaster needs and evolving assessment quality, providing a practical solution for real-world disaster response scenarios where both speed and accuracy are critical. This balance between rapid initial assessment and increasing accuracy represents a significant advancement in disaster response capabilities, enabling emergency managers to make better-informed decisions as more information becomes available while maintaining consistent operational support throughout the response period.

Results

Evaluation framework and dataset construction

We evaluate our framework using two complementary approaches: synthesized data for controlled validation and real-world data for practical performance assessment. The synthesized data, generated from real-world observations, enables verification of the ability of our model to recover complex posteriors under known conditions. To create this validation dataset, we mimic the cascading process of geohazards to DPMs using prior physical models (α_LS, α_LF, α_BD) and actual DPM observations. Specifically, we incorporate three key prior models: the USGS ground failure model for landslides (α_LS)⁶, which uses geospatial factors like slope and lithology combined with ground motion intensities to estimate landslide probabilities; the USGS ground failure model for liquefaction (α_LF)⁷, which evaluates soil susceptibility to liquefaction based on geological conditions and seismic intensities; and the HAZUS building fragility curves (α_BD)⁴⁰, which provide probability distributions of damage states for different building types under various seismic loadings. We first optimize our algorithm to obtain near-optimal posteriors, using flow numbers K = 3, 4, 8 for the 2018 Hokkaido, 2020 Puerto Rico, and 2021 Haiti earthquakes respectively. Ground truth locations are then calibrated to match these posteriors (AUC ~1), creating a controlled test environment.

We run our algorithm with different flow numbers K to identify optimal model configurations. The model exhibiting the best performance (i.e., highest AUC value and tightest variational lower bound with K = k*) is selected as our final approximate posterior distribution. For evaluation purposes, we denote the approximate posterior distribution obtained with different flow numbers K = k as the “test posterior with flow number k". We compare our results against established baselines: the USGS ground failures model^7,41 for landslides and liquefaction, building fragility curves for damage assessment, and the variational causal Bayesian inference method⁴². Additionally, we validate our optimal models using real-world ground truth data^43,44.

Model performance analysis

Effectiveness of noisy-OR structure

The noisy-OR structure in our framework demonstrates superior capability in modeling how multiple hazards independently contribute to observed impacts. For the 2021 Haiti earthquake, where multiple hazards (landslides, liquefaction) independently influence building damage, our model achieves an AUC of 0.9842 for landslide detection while simultaneously maintaining 0.9586 for building damage assessment. This simultaneous high performance across different hazards validates the ability of the noisy-OR structure to capture independent causal influences, unlike traditional approaches that either handle hazards in isolation or assume simple additive effects.

The effectiveness of capturing independent influences is particularly evident in the Puerto Rico case, where landslides and liquefaction occur in distinct geographical regions due to different geological conditions. Our model achieves an AUC of 0.9424 for liquefaction estimation while maintaining high performance for landslide detection (AUC 0.9519), demonstrating its ability to properly attribute signals to their independent causes.

The effectiveness of capturing independent influences is particularly evident in the Puerto Rico case, where landslides and liquefaction occur in distinct geographical regions due to different geological conditions. Our model achieves an AUC of 0.9424 for liquefaction estimation while maintaining high performance for landslide detection (AUC 0.9519), demonstrating its ability to properly attribute signals to their independent causes. Figure 3 visualizes this capability through liquefaction estimation results. Starting from the baseline USGS model (Fig. 3a), our test posteriors with increasing flow numbers (K = 0, 1, 3, 5, 8 shown in Fig. 3b–f) demonstrate progressive improvement in detail and accuracy, ultimately converging to the true posterior (Fig. 3g). This progression demonstrates the ability of our model to capture both independent hazard influences and their complex spatial distributions.

Fig. 3: Visualization of liquefaction probability estimation for the Haiti case.

a Initial estimation from the USGS ground failure model from [7]. b–f Evolution of the estimation of our framework as the number of normalizing flows (K) increases, showing progressive refinement in estimation detail and accuracy. A higher K value allows the model to capture more complex patterns in the liquefaction distribution. g The true posterior distribution derived from validation data. Darker blue indicates a higher probability of liquefaction (scale 0–0.45). The visualization was created using QGIS (QGIS.org, 2024).

Normalizing flow performance

To visualize the effectiveness of our normalizing flow integration, we examine both the DPM reconstructions and the evolution of posterior distributions. Figure S1 (see Supplementary Fig. S1 in Supplementary Information) shows the progression of DPM estimation for the 2021 Haiti earthquake, comparing the real-world DPM (Fig. S1a) generated by the ARIA team⁴⁵ with synthesized DPMs at different flow numbers (K = 0, 3, 8). The results demonstrate progressive improvement in detail and accuracy with increasing flow numbers, ultimately achieving close correspondence with the real-world observations at K = 8.

A detailed examination of the normalizing flow transformations is presented in Fig. S2. Starting from logit-normal initial distributions, the flows transform simple distributions into complex, task-appropriate forms. Figure S2a, e shows the initial distributions for non-landslide and landslide cases respectively. Through successive transformations, these evolve into highly specialized distributions (Fig. S2c, g) that closely match the ground truth posteriors (Fig. S2d, h).

Figure S3a demonstrates how the negative variational lower bound changes with flow length, reaching minimum values at K = 3, 4, and 8 for Hokkaido, Puerto Rico, and Haiti earthquakes respectively. This alignment between optimal flow length and event complexity validates the ability of our model to adapt its expressiveness to scenario requirements.

Case studies across different earthquake events

We evaluate our framework on three earthquake events: the 2021 Haiti (M7.2), 2020 Puerto Rico (M6.4), and 2018 Hokkaido, Japan (M6.6) earthquakes. Figure 4 presents damage proxy maps (DPMs) generated by the ARIA team after these three events, which were used as primary observational data in our analysis. Tables 1 and 2 summarizes the comparative performance of our model against multiple baseline approaches across all events.

Fig. 4: Damage proxy maps (DPMs) generated by the AIRA team⁴⁵ after three events revealing earthquake-induced surface changes across three major seismic events.

a 2021 Haiti earthquake (M7.2), showing extensive surface changes in both urban and mountainous regions; b 2020 Puerto Rico earthquake (M6.4), highlighting concentrated damage patterns in the southern coastal area; c 2018 Hokkaido earthquake (M6.6), demonstrating widespread surface deformation in mountainous terrain. The color intensity indicates the degree of surface change: whiter shades represent more significant changes likely associated with damage, landslides, or ground deformation, while darker shades indicate minimal change. These maps serve as primary observational data for rapid post-disaster assessment, enabling emergency responders to identify severely affected areas before ground surveys become possible.

Table 1 AUC comparison results using real-world data from InSAR imagery

Table 2 F1-score comparison using InSAR data across earthquake events

2021 Haiti earthquake (M7.2)

For the Haiti earthquake, our framework demonstrates robust performance across multiple hazard types, achieving an AUC of 0.9842 for landslide detection and 0.9586 for building damage assessment (Table 1). The narrow confidence intervals (±0.0082 and ±0.0087, respectively) demonstrate the statistical reliability of these improvements over baseline methods, whose wider margins (ranging from ±0.0098 to ±0.0122) suggest greater uncertainty in their estimation. The corresponding F1-scores of 0.9281 for landslide detection and 0.9285 for building damage assessment (Table 2) further validate the effectiveness of our model in both identifying and classifying hazards. Figure S3b, c shows ROC curves comparing our model against baselines. Even without normalizing flows, our posteriors outperform prior models—the test posterior with K = 0 achieves significant improvements over baselines. As we increase the flow numbers, model performance consistently improves, with K = 8 achieving optimal results that closely match the performance of DisasterVINF.

The building damage assessment visualization in Fig. 5 provides detailed insight into model behavior. Figure 5a–d shows the progression of test posteriors with different flow numbers and our DisasterVINF model. When we apply no normalizing flows, our test posterior model identifies major damage areas but lacks detail. As we increase K, the posterior estimates consistently estimate high probabilities in damaged areas while revealing finer details. Figure 5e–h displays these results overlaid with ground truth, demonstrating how the best-performing test posterior (K = 8) achieves performance comparable to the full DisasterVINF model.

Fig. 5: Visualization of building damage estimation in the Haiti case.

Top row a–d Progressive improvement in damage estimation accuracy with increasing normalizing flows (K). Green intensity indicates the estimated probability of building damage (scale 0–1). Bottom row e–h Same estimation overlaid with ground truth data (orange) for validation. Our final DisasterVINF model d, h shows strong agreement with actual damage patterns, particularly in high-density urban areas. Ground truth building damage data from ref. ⁵⁵.

We further validate the performance of our model through DPM comparison. Figure S1 shows DPMs for a focused area of the Haiti earthquake, comparing real-world observations with the estimation of our model at different flow numbers. The synthesized DPMs progressively improve in accuracy and detail, with K = 8 achieving remarkable correspondence with real-world data.

2020 Puerto Rico Earthquake (M6.4)

The Puerto Rico case presents a more complex scenario with multiple interacting hazards. Our model demonstrates significant improvements over baseline approaches, as shown in Table 1. For liquefaction estimation, our model achieves an AUC of 0.9424, significantly outperforming the USGS model (0.8432), ANN model (0.7908), and GBM model (0.7455). The tight confidence interval of our model (±0.0098) compared to baselines (±0.0124 to ±0.0138) further validates the robustness of these improvements. The F1-score analysis shows similar superiority, with our model achieving 0.9154 for liquefaction estimation, substantially outperforming the USGS model (0.8218), ANN model (0.7688), and GBM model (0.7245). Building damage assessment shows similar improvements, with the best test posterior (K = 4) achieving an AUC of 0.9939, representing a 40.98% improvement over the prior AUC value (0.7050).

Figure S4 demonstrates the performance of our model through ROC curves for different hazard types, showing consistent superiority over baseline methods. The achievement of optimal performance at K = 4 suggests that the Puerto Rico event, while complex, requires less model expressiveness than the Haiti case.

2018 Hokkaido earthquake (M6.6)

The Hokkaido earthquake provides an important test case with predominantly landslide hazards in mountainous terrain. Our model achieves an AUC of 0.9508 for landslide detection (Table 1), substantially outperforming the USGS model (0.8565) and other baselines. The F1-score of 0.9192 (Table 2) further demonstrates the superior performance of our model compared to the USGS model (0.8410) and other baselines. The model reaches optimal performance with K = 3, reflecting the relatively simpler causal mechanism compared to multi-hazard scenarios. The ROC curves and AUC values presented in Table S2 and Fig. 6 demonstrate consistent outperformance over baseline models, with progressive improvement as flow numbers increase up to K = 3. This case particularly demonstrates the ability of our model to adapt its complexity to scenario requirements.

Fig. 6

This figure displays the ROC curves for different models in the 2018 Hokkaido earthquake case: the prior landslides models from the USGS, our DisasterVINF model, and test posterior models with different flow numbers.

Comprehensive benchmark comparisons

We conduct extensive comparisons against multiple baseline approaches. For landslide (LS) and liquefaction (LF) estimation, we benchmark against the USGS ground failure models^6,7, VBCI model from ref. ⁴², ANN model from ref. ⁴⁶, and GBM model from ref. ⁴⁶. For building damage (BD) assessment, we compare against the building fragility curve, VBCI model from ref. ⁴², and the ensemble model from ref. ⁴⁷.

Our model demonstrates significant improvements across all hazard types and earthquake events. For the 2021 Haiti earthquake, our landslide detection achieves an AUC of 0.9842, surpassing the USGS model (0.9351), VBCI (0.9412), ANN (0.8413), and GBM (0.8901). The corresponding F1-scores show similar improvements, with our model achieving 0.9281 compared to the USGS model (0.8813), VBCI (0.8929), ANN (0.8126), and GBM (0.8621). Building damage assessment shows similar improvements, with our model achieving an AUC of 0.9586, compared to the building fragility curve (AUC of 0.8120) and VBCI (AUC of 0.9071). In terms of F1 scores, our model achieves 0.9285 for building damage assessment, significantly outperforming the building fragility curve (0.7853) and VBCI (0.8917). In the Puerto Rico case, our liquefaction estimation reaches an AUC of 0.9424, outperforming the USGS model (0.8432), VBCI (0.9077), ANN (0.7908), and GBM (0.7455) by substantial margins. The F1-score analysis further validates these improvements, with our model achieving 0.9154 for liquefaction estimation, considerably higher than all baseline approaches. These improvements are particularly pronounced in multi-hazard scenarios, where traditional single-hazard approaches struggle to distinguish overlapping signals. The benchmark methods, primarily focusing on single hazards through logistic regression models, demonstrate limited capability in capturing the complex interactions that our integrated framework effectively models.

2022 Hurricane Ian

To demonstrate the generalizability of our framework beyond seismic events, we further evaluate the performance of Hurricane Ian, a Category 4 hurricane that made landfall in Florida in September 2022. This evaluation provides an important test case as hurricanes present distinct challenges in terms of their cascading hazard chains—particularly the interaction between wind damage and flood impacts. We also compare our results with state-of-the-art hurricane damage assessment methods on scalable and rapid building damage detection after Hurricane Ian²⁷.

For hurricane scenarios, we adapt our method to capture the physics of hurricane-induced hazards. As shown in Table 3, Online-DisasterVINF demonstrates superior accuracy in assessing hurricane-induced building damage in Lee County, Florida, achieving balanced detection capabilities with a True Positive Rate of 0.8173 and True Negative Rate of 0.7721. The results reveal several important insights about the effectiveness and adaptability of our framework. Our method achieves the highest overall performance with an AUC of 0.7883, surpassing both supervised and unsupervised baseline approaches. This balanced performance is particularly noteworthy when compared to the computer vision models—while FCS-Net⁴⁸ and Dual-HRNet⁴⁹ achieve high TNR values (up to 0.9795), they struggle with TPR, especially without finetuning (as low as 0.0912). This imbalance suggests these models tend to be overly conservative in damage detection when applied to new disaster events. Even after finetuning with Hurricane Ian data, only Dual-HRNet achieves comparable TPR (0.8217) to our method, but at the cost of reduced TNR (0.6251) and requiring labeled training data.

Notably, the performance of our model is significantly better than other label-free approaches—the DPM-based model (AUC = 0.6739) and traditional fragility curve (AUC = 0.5695). This improvement demonstrates the value of our causal modeling approach in capturing the complex interactions between hurricane-induced wind and flood forces. The Bayesian network-based Model, while achieving a slightly higher TPR (0.8293), shows a lower TNR (0.6221), indicating our Online-DisasterVINF framework provides a more balanced and reliable estimation across both damaged and undamaged structures.

Ablation studies and model robustness

Computational efficiency analysis

To demonstrate the suitability of our proposed framework for large-scale applications, we conducted a comprehensive evaluation of its computational complexity and efficiency. Table 4 illustrates processing times across different earthquake events: the Haiti earthquake region (15,879 km²) processes in 6919 s, the larger Puerto Rico region (17,012 km²) in 10,105 s, and the significantly smaller Hokkaido event area (266 km²) in 156 s. These times were achieved using a personal desktop computer with GPU acceleration. Analysis of the variational lower bound reveals that complexity scales approximately linearly with both node degree in the graph and the number of pixels in the map. Our local pruning algorithm further enhances efficiency by making computation sub-linear in overall graph size, achieved by eliminating impossible nodes based on local building footprints and geological conditions. These processing times align well with operational constraints, considering satellite revisiting times typically range from hours to 2 days.

Table 3 Performance comparison of building damage assessment models for hurricane-induced damage detection for Hurricane Ian case study in Lee County, Florida

Table 4 This table shows the time cost of running our framework using the same batch size in three earthquake events using real-world data

Hyperparameter sensitivity

Our comprehensive ablation studies examine the sensitivity of our framework to various hyperparameters. The triplet loss parameter study (λ) in Table 5 reveals optimal performance at λ = 1.0 across all three events, showing significant performance degradation with both smaller (λ = 0.1) and larger (λ = 5.0) values. While this suggests a robust optimal point, we note that these values may vary for different disaster types and scenarios, recommending case-specific tuning for optimal performance.

Table 5 λ in triplet loss evaluation using synthesized data from real InSAR imagery

The sample size analysis presented in Table S3 provides critical insights into Monte Carlo sampling efficiency in our stochastic variational inference. When implementing small sample sizes (M = 10, 100), we observe the significant impact on model accuracy due to inadequate distribution representation. Performance notably stabilizes around M = 5000 samples, with further increases showing diminishing returns despite linear increases in computational cost. This analysis enables users to select appropriate sample sizes balancing accuracy requirements with computational constraints.

Our investigation of weight update sizes (Table S4) demonstrates a clear relationship between step size and model accuracy. Smaller step sizes consistently lead to more accurate results across all event types, though with increased computational time. We observe that performance degrades significantly with larger update sizes, particularly above certain thresholds that vary by event type and scale. This relationship between update size and accuracy provides practical guidance for implementation while highlighting the need for event-specific tuning.

Inference algorithm comparison

We evaluated the effectiveness of our algorithm through a comprehensive comparison of different inference approaches, including VI with local graph pruning, VI full, MCMC local, and MCMC full. Table S5 presents AUC and negative variational lower bound (NVLB) values across these approaches, revealing several key insights. Our pruning-based stochastic VI algorithm consistently outperforms MCMC full in both metrics, achieving this superior performance despite the theoretical guarantee of MCMC of exact marginal probabilities as sampling iterations approach infinity. The similarity between VI full and VI local training values validates the suitability of our stochastic approach for large-scale applications. Furthermore, the decreased AUC and increased NVLB of the MCMC local baseline compared to the full model underscore the effectiveness of our pruning-based approach in maintaining high performance while reducing computational complexity.

Enhancement through online learning

Building upon the capabilities of the base DisasterVINF model, we evaluate how our online learning mechanism improves performance through incremental incorporation of ground truth data. As shown in Table 6 and Fig. 7, we observe consistent performance improvements across all metrics as ground truth integration increases from 0% to 50%. The AUC value shows a steady increase from 0.9586 (initial estimate without ground truth) to 0.9780 (with 50% ground truth), with the most significant improvements occurring in the early stages of data integration. The confidence intervals also narrow progressively from ±0.0113 to ±0.0096, indicating increased estimation reliability with more ground truth data.

Table 6 Model performance metrics with varying ground truth percentages

Fig. 7: Performance evaluation of online-DisasterVINF with incremental ground truth.

a AUC values improvement as the percentage of incorporated ground truth data increases from 0% to 50%. b Comparison of Precision, Recall, and F1 Score metrics across different ground truth percentages. The values of Precision, Recall, and F1 Score metrics are shown in Table 6.

Table 6 provides a detailed breakdown of model performance metrics at different ground truth percentages. The precision value increases from 0.9300 at 0% to 0.9570 at 50% ground truth integration, demonstrating significantly improved reliability in hazard identification. Similarly, recall shows substantial improvement from 0.9250 to 0.9540, indicating enhanced detection coverage across the affected areas. The F1 score, providing a balanced measure of the performance of the model, increases from 0.9275 to 0.9555, confirming overall improvement in both the precision and coverage aspects of the estimation of our model. Notably, the confidence margins for all metrics decrease consistently as more ground truth data is incorporated, with F1-score margins reducing from ±0.0099 to ±0.0087, further validating the increasing reliability of our framework.

These comprehensive results demonstrate several key capabilities of our framework. First, the model provides strong initial estimates even without ground truth data (precision 0.9300, recall 0.9250), crucial for immediate post-disaster response. Second, as shown in Table 6, the incorporation of even limited ground truth data (10–20%) significantly enhances performance, with precision improving to 0.9450 and recall to 0.9420 at 20% ground truth. Finally, the observed diminishing returns pattern in later stages (≥20% ground truth) suggests that our model effectively captures the most important patterns with relatively limited ground truth data, making it particularly suitable for real-world disaster response scenarios where complete ground truth data is rarely available.

Model robustness to noisy ground truth

To evaluate the resilience of our framework to potential errors in field assessments, we conduct systematic experiments using our synthesized dataset from the 2021 Haiti earthquake case. In this section, we examine how noise in ground truth data affects model performance. Using our synthetic data generation process described in Section Evaluation framework and dataset construction, we create controlled test scenarios with varying levels of noise in the ground truth labels. As shown in Table 7, our framework exhibits robust performance even with significant noise in the ground truth data. For building damage assessment, performance degrades gracefully as noise increases—the AUC value decreases by only 1.97% with 5% noise and maintains 90.54% of its baseline performance even with 30% noise in the labels. Landslide detection shows similar robustness, with AUC decreasing by 2.52% at 5% noise and maintaining 89.21% of baseline performance at 30% noise. The F1 scores follow a similar pattern, demonstrating the ability of our framework to maintain both precision and recall despite noisy inputs. These results suggest that our framework can provide reliable estimates even when rapid post-disaster assessments contain significant uncertainties, making it particularly valuable for real-world disaster response scenarios where perfect ground truth data are rarely available.

Table 7 Model performance with noisy ground truth data using the synthetic validation dataset

Discussion

In this work, we establish a novel approach to modeling the complex interactions of multiple hazards and impacts during disasters, as revealed by otherwise difficult-to-decouple sensing signals. Our framework, online-DisasterVINF, is co-designed with existing USGS Ground Failure products and the PAGER system to provide rapid, accurate, and high-resolution estimates of multiple hazards and impacts. We utilize SAR data, renowned for its resilience to cloud cover due to its operation in the microwave region of the electromagnetic spectrum. This feature allows for comprehensive spatial coverage and high-resolution information about ground surface changes across various weather conditions⁵⁰.

A key innovation in our work is the use of Noisy-OR structures, which are prevalent in physical models, to efficiently capture the complex interdependencies among multiple hazards and impacts. This approach, combined with the introduction of normalizing flows, effectively models the complex causal dependencies in cascading disaster networks. The integration of a triplet loss leveraging prior physical knowledge improves the identifiability of our highly expressive networks. Normalizing flows accurately capture the non-linear relationships between hazards and impacts, enhancing the interpretability of the developed framework⁵¹.

Our comprehensive evaluation of three major earthquakes demonstrates the robustness and adaptability of our model. For the 2021 Haiti earthquake (M7.2), our model achieved an AUC of 0.9842 for landslide detection and 0.9586 for building damage assessment. The framework showed particular strength in the Puerto Rico case (M6.4), achieving an AUC of 0.9424 for liquefaction estimation—an 11.76% improvement over prior methods. The architecture of our framework can be adapted to model other interacting natural hazards beyond earthquake scenarios. For instance, in earthquake-triggered tsunami events, our causal graph structure could capture the sequential nature of hazards, with ground shaking as the initial trigger leading to tsunami generation and subsequent coastal infrastructure damage. The key components would maintain their roles while adapting to new hazard characteristics: prior models would incorporate hazard-specific parameters (e.g., wave height and inundation depth for tsunamis)^52,53, normalizing flows would model the complex relationships between multiple hazard intensities and structural performance, and the damage nodes would be expanded to represent different failure modes. This adaptability is particularly valuable for compound hazard scenarios where multiple threats interact to cause damage. We also validate the generalizability of our model by evaluating our model on the 2022 Hurricane Ian. Our model achieves an AUC of 0.7883 and demonstrated balanced detection capabilities with a TPR of 0.8173 and TNR of 0.7721, significantly outperforming baseline models. Notably, while computer vision models like FCS-Net and Dual-HRNet achieved high TNR values, they struggled with TPR without extensive finetuning, highlighting the advantage of our model in rapid deployment without requiring labeled training data. For computational efficiency, our pruning strategy enables the processing of large-scale disasters, handling areas up to 15,879 km² within 2–120 min while maintaining accuracy.

Another significant advancement is the ability of our framework to perform online updating, incorporating varying percentages of ground truth data as it becomes available. This causal representation learning with partial ground truth allows online-DisasterVINF to incrementally improve estimation in real time as more verified information is gathered in post-disaster scenarios, bridging the gap between initial rapid assessments and more refined analyses. online-DisasterVINF demonstrates significant performance improvements compared to state-of-the-art benchmark models, including those used by existing USGS products. Our analysis reveals substantial improvements in model performance, with AUC scores increasing from 0.9586 to 0.9780 as ground truth data integration increases from 0% to 50%, underlining the adaptability and potential for real-time refinement of our framework in disaster response applications.

The significance of our framework extends beyond its immediate applications in earthquake scenarios. The noisy-OR structure provides a principled way to model-independent causal influences in disaster chains, while the normalizing flows capture complex non-linear relationships that simple log-linear models cannot represent. This combination offers a new paradigm for disaster modeling that is both physically interpretable and mathematically expressive. Our online learning mechanism demonstrates that accurate initial estimates can be achieved without ground truth data while maintaining the flexibility to incorporate field observations as they become available—a crucial capability for real-world disaster response.

While these results are promising, we acknowledge several important limitations of our current framework. First, the performance of online-DisasterVINF depends on the quality of prior physical models, which may have their own inherent uncertainties and limitations. For example, the USGS ground failure models we use as priors may not fully capture all local geological conditions that influence hazard susceptibility. Second, while our framework can operate without ground truth data initially, the accuracy of refined estimates still relies on the representativeness of available field observations. In regions where reconnaissance data is sparse or biased towards certain areas, the updates of online-DisasterVINF may not uniformly improve predictions across all locations. Third, our current implementation assumes the independence between certain types of hazards, which may not fully capture all possible interaction effects in complex disaster scenarios.

This flexibility in modeling different types of hazard chains—from cascading seismic hazards to compound flood-wind effects—demonstrates how our approach of combining physics-based priors with causal modeling can be adapted for various disaster types while maintaining rapid assessment capabilities. However, the adaptation to new hazard types requires careful consideration of appropriate prior models and potentially modified causal graph structures to reflect different physical relationships. Future work could explore automated methods for learning optimal graph structures from data, as well as incorporating additional data sources such as social media information or crowdsourced damage reports⁵⁴.

The performance of our framework in both earthquake and hurricane scenarios suggests its potential applicability for different types of rapid disaster assessment. While our current results focus primarily on building damage and ground failures, future research could expand the framework to assess other critical infrastructure systems and socioeconomic impacts. The ability of our model to handle both the initial rapid assessment phase and subsequent refinement phases, combined with its computational efficiency in processing large-scale disasters, addresses an important need in current disaster response systems.