WGFINNs: Weak formulation-based GENERIC formalism informed neural networks

Jun Sur Richard Park; Auroni Huque Hashim; Siu Wun Cheung; Youngsoo Choi; Yeonjong Shin

cs.LGApr 3, 2026

WGFINNs: Weak formulation-based GENERIC formalism informed neural networks

Authors: Jun Sur Richard Park, Auroni Huque Hashim, Siu Wun Cheung, Youngsoo Choi, Yeonjong Shin

Organizations: Division of Big Data Science, Korea University, Sejong, 30019, Republic of Korea · Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA · Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA.

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 & tables

Fig. 1 : Example 5.3 . Comparison of weighted and unweighted loss functions for noise-free data. Predictions by GFINNs and WGFINNs for all state variables.

Fig. 2 : Example 5.3 . Ground-truth (GT) trajectories and the corresponding predictions by WGFINNs and GFINNs with 5 $\%$ -noise corruption.

Panel of Figure 3 — Fig. 3 : (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 ( 11 ) at $\sigma=10^{-2}$ . There exists a stepsize that yields the exact estimation of the strong form solution.

Panel of Figure 4 — Fig. 4 : Left: The relative errors of the weak form solutions with respect to the length of the support $S$ using the test function from Theorem 5 , 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.

Fig. 5 : Example 5.1 . The relative $\ell_{2}$ test errors ( 14 ) under varying noise levels. The solid line represents the mean across five independent simulations.

Panel of Figure 6 — Fig. 6 : Example 5.1 . Ground-truth (GT) trajectories and the corresponding predictions by WGFINNs and GFINNs. From top to bottom: noise-free data, 5 $\%$ noise, and 10 $\%$ noise. For each method, the results obtained from the model with the smallest training loss among five independent runs are shown.

Panel of Figure 7 — Fig. 7 : Example 5.1 . Learned energy and entropy: (Top) Energy contours in the $q$ - $p$ plane ( $S_{1}=S_{2}=2.5$ fixed). (Middle) Entropy contours in the $S_{1}$ - $S_{2}$ plane ( $q=1,p=0$ fixed). (Bottom) Calibrated entropy $S_{\mathrm{NN}}$ along a predicted trajectory for noise levels $0\%$ , $5\%$ , and $10\%$ . All learned quantities are calibrated via least squares.

Fig. 8 : Example 5.2 . The relative $\ell_{2}$ test errors ( 14 ) under varying noise levels. The solid line represents the mean across five independent simulations.

Panel of Figure 9 — Fig. 9 : Example 5.2 . Ground-truth (GT) trajectories and the corresponding predictions by WGFINNs and GFINNs. From top to bottom: noise-free data, 5 $\%$ noise, and 10 $\%$ noise. For each method, the results obtained from the model with the smallest training loss among five independent runs are shown.

Fig. 10 : Example 5.3 . The relative $\ell_{2}$ test errors ( 14 ) under varying noise levels. The solid line represents the mean across five independent simulations.

Panel of Figure 11 — Fig. 11 : Example 5.3 . GT trajectories and the corresponding predictions by WGFINNs and GFINNs. (Top) noise-free, i.e., 0 $\%$ noise and (Bottom) 10 $\%$ noise. See Figure 2 for 5 $\%$ noise. For each method, the results obtained from the model with the smallest training loss among five independent runs are shown.

Fig. 12 : Example 5.4 . The relative $\ell_{2}$ test errors ( 14 ) with respect to noise levels. The solid line represents the mean across five independent simulations.

Fig. 13 : Example 5.4 . GT states $E_{1}$ , $E_{2}$ , and the corresponding predictions by WGFINNs and GFINNs. From left to right: $0\%$ noise (noise-free), $3\%$ noise, and $5\%$ noise. For each method, the results obtained from the model with the median training loss among five independent runs are shown.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

WGFINNs
Noise	LR	Decay rate	Decay steps	$\ell/p/s$	$\eta^{*}$ (RBA)	$\gamma$ (RBA)
$0\%$	1e-3	0.99	1000	30/6/0.9	0.005	0.995
$5\%$	1e-3	0.25	15000	80/4/0.9	0.01	0.99
$10\%$	1e-3	0.1	15000	80/4/0.9	0.005	0.995
GFINNs
Noise	LR	Decay rate	Decay steps	Batch size	$\eta^{*}$ (RBA)	$\gamma$ (RBA)

Appendix

Table 1: Example 5.1 : Hyperparameters for WGFINNs and GFINNs at varying noise levels. The decay rate

\alpha

means the multiplication by

\alpha

is applied to the current learning rate.

WGFINNs
Noise	LR	Decay rate	Decay steps	$\ell/p/s$	$\eta^{*}$ (RBA)	$\gamma$ (RBA)
$0\%$	1e-3	0.99	1000	10/4/0.7	0.0	1
$5\%$	1e-3	0.1	7500	40/2/0.95	0.01	0.99
$10\%$	1e-3	0.1	7500	40/2/0.95	0.05	0.95
GFINNs
Noise	LR	Decay rate	Decay steps	Batch size	$\eta^{*}$ (RBA)	$\gamma$ (RBA)

Appendix

Table 2: Example 5.2 : Hyperparameters for WGFINNs and GFINNs at varying noise levels. The decay rate

\alpha

means the multiplication by

\alpha

is applied to the current learning rate.

WGFINNs
Noise	LR	Decay rate	Decay steps	$\ell/p/s$	$\eta^{*}$ (RBA)	$\gamma$ (RBA)
$0\%$	1e-3	0.95	1000	60/4/0.8	0.0	1.0
$5\%$	1e-4	0.95	1000	60/4/0.8	0.0	1.0
$10\%$	1e-4	0.85	1000	60/4/0.8	0.0	1.0
GFINNs
Noise	LR	Decay rate	Decay steps	Batch size	$\eta^{*}$ (RBA)	$\gamma$ (RBA)

Appendix

Table 3: Example 5.3 : Hyperparameters for WGFINNs and GFINNs at varying noise levels. The decay rate

\alpha

means the multiplication by

\alpha

is applied to the current learning rate.

WGFINNs
Noise	LR	Decay rate	Decay steps	$\ell/p/s$	$\eta^{*}$ (RBA)	$\gamma$ (RBA)
$0\%$	1e-3	0.98	500	10/3/0.9	0.0	1.0
$3\%$	1e-3	0.95	1000	20/3/0.9	0.005	0.995
$5\%$	1e-3	0.95	1000	20/3/0.9	0.05	0.95
GFINNs
Noise	LR	Decay rate	Decay steps	Batch size	$\eta^{*}$ (RBA)	$\gamma$ (RBA)

Appendix

Table 4: Example 5.4 : Hyperparameters for WGFINNs and GFINNs at varying noise levels. The decay rate

\alpha

means the multiplication by

\alpha

is applied to the current learning rate.

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