Robust learning from sparse and noisy data for model discovery of nonlinear systems

Discovering governing equations from sparse, noisy observational data remains a fundamental challenge in data-driven science. We present Deep Learning-Enhanced Automatic Model Discovery (DL-AMD), a decoupled framework that separates data reconstruction from model identification through two distinct stages: the first employs residual attention neural networks for mesh-free reconstruction and implicit denoising; the second performs automatic differentiation followed by adaptive sparse regression with information-theoretic model selection. Unlike SINDy-based methods where finite difference derivatives amplify measurement noise, DL-AMD obtains robust derivatives through neural network interpolation. In contrast to physics-informed approaches that couple data fitting with physics constraints—leading to gradient conflicts and prohibitive computational costs—our architecture solves independent optimization subproblems, reducing complexity from multiplicative to additive scaling. Validation on six benchmarks (Cubic, Duffing, Van der Pol ODEs; Burgers, Allen-Cahn, Navier–Stokes PDEs) demonstrates 2–6× speedup with comparable or superior accuracy under 10%–15% noise and 10%–30% sampling. Source code is available at https://github.com/xgxgnpu/DL_AMD.