WGFINNs: Weak formulation-based GENERIC formalism informed neural networks

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.

• The paper introduces a novel neural network framework that integrates weak formulation with GENERIC constraints to enhance noise-robust discovery of governing equations.
• Its methodology employs state-wise weighted loss and residual-based attention to ensure thermodynamic consistency and improve prediction accuracy under noise.
• Empirical results and theoretical analysis confirm that WGFINNs outperform strong-form approaches, maintaining conservation laws even with noisy measurements.

Weak Formulation-Based GENERIC Formalism Informed Neural Networks (WGFINNs): A Technical Review

Introduction

This work introduces Weak formulation-based GENERIC formalism informed neural networks (WGFINNs), a methodological advance for data-driven discovery of governing equations in physical systems. The framework specifically addresses the challenge of learning robust, physically-constrained models from noisy observational data. By integrating the GENERIC (General Equation for the Non-Equilibrium Reversible–Irreversible Coupling) thermodynamic formalism with weak-form loss functions and architectural modifications, WGFINNs achieve improved noise robustness, while maintaining thermodynamic consistency and interpretability in learned dynamics.

Theoretical Background and Motivation

GFINNs (GENERIC formalism informed neural networks) [Zhang et al., Phil. Trans. Roy. Soc. A, 2022] leverage the GENERIC structure to guarantee adherence to non-equilibrium thermodynamics—encoding reversibility, irreversibility, symmetry, and degeneracy properties directly into the neural architecture. However, the original GFINN framework uses a strong-form loss based on pointwise residuals of governing equations, rendering it highly susceptible to noise when computing temporal and spatial derivatives.

This limitation motivates a transition to weak-form methodologies, well-recognized for their noise robustness. In the weak-formulation, the PDE residuals are integrated after being multiplied by test functions, which enables the estimation of derivatives in an integral sense, attenuating noise amplification. Theoretical analysis in this work establishes that, in the presence of noisy measurements, strong-form estimators diverge as the discretization step shrinks, while the weak-form estimator can remain accurate under appropriate test function choices.

Methodology

The WGFINN architecture synthesizes several innovations:

• Weak-form loss integration: The governing laws are enforced by integrating the weak form of the GENERIC equations with judiciously chosen test functions, rather than enforcing pointwise residual minimization.
• State-wise weighted loss and residual-based attention: Scale imbalances among state variables are mitigated via a state-adaptive weighted loss and a residual-based attention mechanism, the latter informed by recent advances in attention for PINNs [Anagnostopoulos et al., CMAME, 2024]. This enhances both optimization stability and accuracy.
• Exact satisfaction of GENERIC constraints: The neural parameterization is constructed so as to preserve the degeneracy and symmetry conditions inherent to the GENERIC formalism, guaranteeing thermodynamic admissibility of the learned dynamics at all times.

The network accepts noisy time-series data for observed state variables, projecting them through a neural operator constrained by GENERIC structure and trained with the weak-form loss.

Theoretical Contributions

A quantitative theoretical comparison between strong- and weak-form estimators is established. For additive noise, the mean-squared error of strong-form estimators grows unboundedly as discretization becomes finer—contrastingly, the weak-form estimator's error is bounded and can be driven to zero even with fixed noise, under regularity and test function admissibility. This analysis formalizes empirical observations that weak-form identification is inherently more robust to measurement imprecision.

Furthermore, the architecture ensures invariance under discretization, as the weak-form naturally accommodates varying mesh resolutions and nonuniform sample distributions.

Numerical Experiments and Results

Numerical studies encompass several dynamical systems governed by GENERIC structure, evaluated across a range of synthetic noise amplitudes. In all cases, WGFINNs exhibit:

• Superior prediction accuracy and physical quantity recovery relative to strong-form GFINNs, with a significant gap at moderate to high noise levels.
• Insensitivity of solution quality to noise when leveraging the weak-form, as opposed to rapid degradation in the GFINN baseline.
• Preservation of conservation laws and thermodynamic constraints throughout training and inference.

Quantitative evaluations report lower prediction errors and improved recovery of energy, entropy, and other physical invariants, regardless of noise strength—a marked improvement over prior approaches.

Implications and Future Directions

WGFINNs advance scientific machine learning in several dimensions. First, they provide a principled framework for learning physics-constrained models from realistic, noisy measurements—addressing a ubiquitous practical bottleneck. Second, the theoretical linkage between GENERIC constraints and weak-form regression sets a precedent for future integration of structure-preserving principles and noise-robust identification methods.

Practical implications include more reliable system identification for experimental datasets typical in engineering and physical sciences, where high-fidelity, low-noise data are scarce. The work also opens avenues for extending weak-form, thermodynamics-consistent neural modeling to multiscale, stochastic, or spatially extended systems, and potential integration with latent variable modeling for high-dimensional data regimes.

Future developments might include exploration of test function optimization strategies, generalization to generative modeling within the GENERIC framework, and scalable algorithms for parameter inference in large-scale physical systems.

Conclusion

WGFINNs represent a substantial methodological contribution for interpretable, noise-robust discovery of governing equations in physical systems, combining the strengths of weak-form regression and the principled constraints of the GENERIC formalism. Empirical and theoretical results confirm significant improvements in accuracy and physical consistency under noisy data, with implications for both scientific machine learning theory and application to real-world dynamical modeling.