Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems

TL;DR

This work tackles the challenging problem of discovering governing PDEs for nonlinear dynamic systems from noisy measurements, where high correlation among candidate terms and stiff/high-order dynamics impede traditional sparse identification. It introduces a physics-informed deep-learning framework that fuses analytical B-spline differentiation with sequential denoising (SRDD), Uncorrelated Component Analysis (UCA), and physics-informed spline fitting (PISF) to iteratively prune candidate terms and recover the governing equation and its parameters. The approach demonstrates robust performance across a suite of ODEs/PDEs (including Van der Pol, Duffing, KS, Burgers, KDV, Navier–Stokes, and 2D wave equations) at up to 10% measurement noise, achieving accurate equation forms and low coefficient of variation in parameter estimates. By jointly fitting data and physics within a unified DL architecture, the method offers a practical, interpretable pathway for data-driven model discovery in civil, mechanical, and fluid dynamics, with future potential for coupled systems and broader applicability.

Abstract

Sparse system identification of nonlinear dynamic systems is still challenging, especially for stiff and high-order differential equations for noisy measurement data. The use of highly correlated functions makes distinguishing between true and false functions difficult, which limits the choice of functions. In this study, an equation discovery method has been proposed to tackle these problems. The key elements include a) use of B-splines for data fitting to get analytical derivatives superior to numerical derivatives, b) sequentially regularized derivatives for denoising (SRDD) algorithm, highly effective in removing noise from signal without system information loss, c) uncorrelated component analysis (UCA) algorithm that identifies and eliminates highly correlated functions while retaining the true functions, and d) physics-informed spline fitting (PISF) where the spline fitting is updated gradually while satisfying the governing equation with a dictionary of candidate functions to converge to the correct equation sequentially. The complete framework is built on a unified deep-learning architecture that eases the optimization process. The proposed method is demonstrated to discover various differential equations at various noise levels, including three-dimensional, fourth-order, and stiff equations. The parameter estimation converges accurately to the true values with a small coefficient of variation, suggesting robustness to the noise.

Figures (24)

• Figure 1: Proposed framework for equation discovery of nonlinear dynamic systems. (A) The measured noisy field data from a general two-dimensional dynamic system at discrete points (B) Approximation of the measured data using the linear combination of B-spline basis functions with unknown coefficients $\bm{\beta}$. (C) Generation of the library comprising of various terms via the fitted B-spline functions. The spatial and temporal derivatives are analytically derived using the B-spline functions. (D) The underlying governing equation $h(x,t)$ with unknown parameters $\bm{\theta}$ the dynamic system follows, which needs to be discovered. (E) The global optimization function for equation discovery requires the simultaneous approximation of the measured data with B-spline functions to obtain $\bm{\beta^*}$, satisfying the governing equation utilizing the dictionary of candidate functions, and sparse estimation of the corresponding parameters $\bm{\theta^*}$.
• Figure 2: A brief description of challenges in sparse system identification and proposed solution strategy. (A) The measured noise data is input to the deep learning network that passes through an encoder and provides scaling factors $\bm{\beta}$ for B-spline fitting (B) The approximation of measured data using B-spline functions and $\bm{\beta}$ obtained from A (C) The setup of the regression formulation using the chosen library of functions for equation discovery. The noise and highly correlated functions pose a challenge to sparse system identification (E1) Sparse regression using L1-regularization to remove small contribution functions (E2) Uncorrelated Component Analysis to remove highly correlated functions that provide repetitive information (E3) Physics-informed spline fitting for simultaneous data fitting and equation discovery.
• Figure 3: Deep learning architecture
• Figure 4: Spline fitting to the measured data from Van der Pol oscillator, analytically derived first derivative and a closeup look from 6-9 seconds.
• Figure 5: Spline fitting and its first derivative with a) no regularization, b) regularization till 2nd derivative, and c) regularization till 4th derivative.