Physics-Informed Neural Networks for Solving Differential Equations

Physics-Informed Neural Networks (PINNs) augment traditional neural architectures by embedding the governing equations of physical systems directly into the loss function. Instead of solely minimising data misfit, PINNs impose a residual term derived from differential operators, ensuring that solutions satisfy the underlying Partial Differential Equations (PDEs) or Ordinary Differential Equations (ODEs). This approach eliminates the need for domain discretisation and meshing, thereby offering a mesh-free and flexible framework for forward and inverse problems. PINNs operate by sampling collocation points in the domain, evaluating network predictions and their derivatives via automatic differentiation, and optimising network parameters to reduce both data error and physics residuals simultaneously. The resulting models can interpolate sparse measurements, infer hidden states or parameters, and deliver continuous, differentiable approximations across complex geometries. Recent advances have addressed scalability to high-dimensional problems, improved training stability through adaptive weighting of loss components, and integrated multimodal data for robust model calibration. The global impact spans fluid mechanics, materials science, geophysics and biomedical engineering, with concrete examples including rapid simulation of turbulent flows, inverse parameter estimation in nanophotonics and accelerated cardiac electrophysiology mapping.

Research from Nature Portfolio

Recent studies have demonstrated the use of sparse regression techniques in conjunction with PINNs to discover closed-form governing equations from limited and noisy data. By coupling deep neural representations with automatic computation of derivatives and a sparsity-promoting regression algorithm, researchers have shown that principal derivative terms and their coefficients can be identified reliably, even under severe data scarcity. This framework has been validated on a variety of nonlinear spatiotemporal systems under different boundary conditions, revealing its potential for automated model discovery in contexts where traditional data-driven or purely analytical approaches are infeasible.

Physics-Informed Neural Networks for Solving Differential Equations publication trend

The graph below shows the total number of articles in physics-informed neural networks for solving differential equations across all publications each year (not limited to Nature Index journals).

Technical terms

Physics-Informed Neural Network (PINN): A neural network that embeds differential equations governing a physical system into its training loss, enforcing consistency with known laws.

Partial Differential Equation (PDE): An equation involving derivatives with respect to multiple independent variables, describing spatial and temporal changes in physical systems.

Residual: The discrepancy at a collocation point between the neural network’s prediction and the value that exactly satisfies the differential equation.

Automatic Differentiation: A technique for computing exact derivatives of functions implemented as computer code, crucial for evaluating differential operator terms in PINNs.

Collocation Point: A sample location in the problem domain where the governing equations are evaluated and enforced during training.

Inverse Problem: A class of problems where unknown parameters or states are inferred from observed data rather than directly prescribed.

References

1. Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next. Journal of Scientific Computing (2022).
2. Physics-informed learning of governing equations from scarce data. Nature Communications (2021).
3. Physics-informed neural networks for inverse problems in nano-optics and metamaterials.. Optics Express (2020).
4. Physics-informed neural networks for solving Reynolds-averaged Navier–Stokes equations. Physics of Fluids (2022).
5. Physics-Informed Neural Networks for Cardiac Activation Mapping. Frontiers in Physics (2020).