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.

Figures (13)

• Figure 2.1: Example \ref{['sec:linearly_damped']}. Comparison of weighted and unweighted loss functions for noise-free data. Predictions by GFINNs and WGFINNs for all state variables.
• Figure 2.2: Example \ref{['sec:linearly_damped']}. Ground-truth (GT) trajectories and the corresponding predictions by WGFINNs and GFINNs with 5$\%$-noise corruption.
• Figure 3.1: (Left) The relative errors of the strong form solution with respect to the step size $\Delta t$ at varying noise levels $\sigma$. The target parameter is $\lambda = -2$ and the noises are drawn from the Gaussian distributions. At each $\Delta t$ and $\sigma$, the scattered marks correspond to 10 independent numerical simulations. The solid lines represent the mean errors over 1,000 independent simulations. The dashed-line is the numerical integration error. (Right) The stepsize multiplied absolute errors with respect to the stepsize. The dashed-line is the limit value from \ref{['eqn:thm-strong-limit']} at $\sigma=10^{-2}$. There exists a stepsize that yields the exact estimation of the strong form solution.
• Figure 3.2: Left: The relative errors of the weak form solutions with respect to the length of the support $S$ using the test function from Theorem \ref{['prop:weak-form-L1']}, at varying different noise levels $\sigma$. Right: The relative errors of both the strong and weak form solutions with respect to the noise levels. In both figures, the scattered marks correspond to 10 independent numerical simulations at each fixed setup. The solid lines are the mean errors over 1,000 independent simulations.
• Figure 4.1: Example \ref{['sec:gas_containers']}. The relative $\ell_2$ test errors (\ref{['eq:test_rmse']}) under varying noise levels. The solid line represents the mean across five independent simulations.

Theorems & Definitions (11)

• Proposition 1.1
• proof
• Theorem 1.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Lemma 1.1