WGFINNs: Weak formulation-based GENERIC formalism informed neural networks

TL;DR

WGFINNs introduce a weak formulation approach to GENERIC-informed neural networks, significantly improving robustness to noisy data while preserving physical laws in scientific machine learning.

Contribution

The paper proposes WGFINNs, a novel weak formulation-based method that enhances noise robustness in GENERIC-informed neural networks without sacrificing physical constraints.

Findings

WGFINNs outperform GFINNs across various noise levels.

Weak-form estimators remain accurate with noisy data under certain conditions.

Numerical experiments show improved prediction accuracy and physical quantity recovery.

Abstract

Data-driven discovery of governing equations from noisy observations remains a fundamental challenge in scientific machine learning. While GENERIC formalism informed neural networks (GFINNs) provide a principled framework that enforces the laws of thermodynamics by construction, their reliance on strong-form loss formulations makes them highly sensitive to measurement noise. To address this limitation, we propose weak formulation-based GENERIC formalism informed neural networks (WGFINNs), which integrate the weak formulation of dynamical systems with the structure-preserving architecture of GFINNs. WGFINNs significantly enhance robustness to noisy data while retaining exact satisfaction of GENERIC degeneracy and symmetry conditions. We further incorporate a state-wise weighted loss and a residual-based attention mechanism to mitigate scale imbalance across state variables. Theoretical analysis contrasts quantitative differences between the strong-form and the weak-form estimators. Mainly, the strong-form estimator diverges as the time step decreases in the presence of noise, while the weak-form estimator can be accurate even with noisy data if test functions satisfy certain conditions. Numerical experiments demonstrate that WGFINNs consistently outperform GFINNs at varying noise levels, achieving more accurate predictions and reliable recovery of physical quantities.