Fig. 1

Overview of the IRK-SINDy framework: (a) For each benchmark problem, we perform measurements that incorporate noise and, thereafter form a dataset. Our objective is to construct a model that is parsimonious, interpretable, and possesses generalizability, capable of accurately forecasting reference dynamics. (b) Given an appropriate initial guess (e.g., \(X(t_{k})\)), the stage values of the IRKs are approximated by solving the system of nonlinear equations (6b) through iterative schemes. In this context, we employ two iterative approaches: (i) fixed point iteration and (ii) Newton’s method. (c) With the stage values established the subsequent step values are computed according to eq. (8). This computational process is depicted as the systematic IRK network. (d) Within this structured representation of IRK-SINDy, the dataset is classified into two categories: forward and backward, followed by the formation of a symbolic features library comprising candidate nonlinear functions. To solve a nonlinear sparse regression problem using the forward and backward predictions illustrated in (b) and (c), an IRK step is applied, and the loss function is minimized by choosing a suitable optimizer. Following a certain number of epochs, a sparsity-promoting algorithm is employed. Finally, every non-zero element in the coefficient matrix \(\xi ^{*}\) signifies an active term within the feature library, thereby representing the resultant discovered model.