Unlearning Noise in PINNs: A Selective Pruning Framework for PDE Inverse Problems

Yongsheng Chen; Yong Chen; Wei Guo; Xinghui Zhong

cs.LGFeb 23, 2026

Unlearning Noise in PINNs: A Selective Pruning Framework for PDE Inverse Problems

Authors: Yongsheng Chen, Yong Chen, Wei Guo, Xinghui Zhong

Organizations: School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. · Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania, Philadelphia, PA 19104, USA. · Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 70409, USA.

Abstract

Physics-informed neural networks (PINNs) provide a promising framework for solving inverse problems governed by partial differential equations (PDEs) by integrating observational data and physical constraints in a unified optimization objective. However, the ill-posed nature of PDE inverse problems makes them highly sensitive to noise. Even a small fraction of corrupted observations can distort internal neural representations, severely impairing accuracy and destabilizing training. Motivated by recent advances in machine unlearning and structured network pruning, we propose P-PINN, a selective pruning framework designed to unlearn the influence of corrupted data in a pretrained PINN. Specifically, starting from a PINN trained on the full dataset, P-PINN evaluates a joint residual--data fidelity indicator, a weighted combination of data misfit and PDE residuals, to partition the training set into reliable and corrupted subsets. Next, we introduce a bias-based neuron importance measure that quantifies directional activation discrepancies between the two subsets, identifying neurons whose representations are predominantly driven by corrupted samples. Building on this, an iterative pruning strategy then removes noise-sensitive neurons layer by layer. The resulting pruned network is fine-tuned on the reliable data subject to the original PDE constraints, acting as a lightweight post-processing stage rather than a complete retraining. Numerical experiments on extensive PDE inverse-problem benchmarks demonstrate that P-PINN substantially improves robustness, accuracy, and training stability under noisy conditions, achieving up to a 96.6% reduction in relative error compared with baseline PINNs. These results indicate that activation-level post hoc pruning is a promising mechanism for enhancing the reliability of physics-informed learning in noise-contaminated settings.

Figures & tables

Figure 1: Schematic representation of the PINN formulation used in this work.

Panel of Figure 2 — Figure 2: P-PINN workflow and a Navier–Stokes inverse example. Left: schematic of the P-PINN pipeline. Right: Navier–Stokes inversion: A baseline PINN is trained on noisy space–time observations; the observations are then partitioned into $\mathcal{D}_{\text{retain}}$ (blue) and $\mathcal{D}_{\text{forget}}$ (red) using residual–data-fidelity scores. The network is iteratively pruned and selectively fine-tuned on $\mathcal{D}_{\text{retain}}$ to obtain P-PINN. Color maps compare the baseline and P-PINN reconstructions of the pressure field $p(x,y,t{=}1)$ with the reference solution.

Problem	$\mathbf{N}$	$\mathbf{N_{PDE}}$	$\mathbf{N_{BC/IC}}$	Noise ( $\sigma_{high}/\sigma_{low}$ , Split)
Parameter Inversion
PInv	$2{,}500$	$8{,}192$	$2{,}048/-$	$1.0/0.1$ ( $60\%/40\%$ )
HInv	$2{,}500$	$8{,}192$	$2{,}048/2{,}048$	$1.0/0.1$ ( $60\%/40\%$ )
NSInv	$10{,}000$	$2{,}000$	–	$\sqrt{0.5}/0.1$ ( $60\%/40\%$ )
EBInv	$10{,}000$	$2{,}000$	$100/200$	$1.0/0.1$ ( $60\%/40\%$ )
WInv	$10{,}000$	$2{,}000$	$100/200$	$1.0/0.1$ ( $60\%/40\%$ )

Table 1: Summary of problem configurations. Here

N

denotes the number of noisy observations,

N_{\mathrm{PDE}}

the number of interior collocation points for enforcing the PDE residual, and

N_{\mathrm{BC/IC}}

the numbers of boundary and initial collocation points (reported as

N_{\mathrm{BC}}/N_{\mathrm{IC}}

, with “–” indicating that a term is not present). The Noise column specifies the standard deviation for high/low corruption subsets and their respective proportions (

\sigma_{high}/\sigma_{low}

, Split %).

L2RE	Data assimilation				Parameter inversion
	Poisson	Heat	Wave	Stokes	PInv	HInv	NSInv	EBInv	WInv
PINN	$2.59\mathrm{E}{-}1$	$9.63\mathrm{E}{-}1$	$7.66\mathrm{E}{-}1$	$6.68\mathrm{E}{-}1$	$9.37\mathrm{E}{-}2$	$1.32\mathrm{E}{-}1$	$3.44\mathrm{E}{+}0$	$2.50\mathrm{E}{-}1$	$9.94\mathrm{E}{-}2$
PINN	$\pm 1.34\mathrm{E}{-}2$	$\pm 2.19\mathrm{E}{-}1$	$\pm 6.81\mathrm{E}{-}2$	$\pm 6.50\mathrm{E}{-}3$	$\pm 2.44\mathrm{E}{-}2$	$\pm 5.78\mathrm{E}{-}3$	$\pm 1.09\mathrm{E}{+}0$	$\pm 1.96\mathrm{E}{-}2$	$\pm 2.60\mathrm{E}{-}3$
P-PINN	$\mathbf{8.04\mathrm{E}{-}2}$	$\mathbf{3.26\mathrm{E}{-}2}$	$\mathbf{8.88\mathrm{E}{-}2}$	$\mathbf{1.19\mathrm{E}{-}1}$	$\mathbf{1.82\mathrm{E}{-}2}$	$\mathbf{8.14\mathrm{E}{-}2}$	$6.44\mathrm{E}{-}1$	$\mathbf{6.95\mathrm{E}{-}2}$	$\mathbf{5.71\mathrm{E}{-}2}$
P-PINN	$\mathbf{\pm 3.13\mathrm{E}{-}2}$	$\mathbf{\pm 1.56\mathrm{E}{-}4}$	$\mathbf{\pm 4.38\mathrm{E}{-}4}$	$\mathbf{\pm 2.03\mathrm{E}{-}2}$	$\mathbf{\pm 2.17\mathrm{E}{-}3}$	$\mathbf{\pm 3.86\mathrm{E}{-}3}$	$\pm 2.67\mathrm{E}{-}1$	$\mathbf{\pm 7.14\mathrm{E}{-}3}$	$\mathbf{\pm 3.46\mathrm{E}{-}3}$
B-PINN	$3.67\mathrm{E}{-}1$	$3.94\mathrm{E}{-}1$	$6.97\mathrm{E}{-}1$	$6.95\mathrm{E}{-}1$	$5.70\mathrm{E}{-}1$	$2.68\mathrm{E}{-}1$	$1.98\mathrm{E}{+}0$	$7.82\mathrm{E}{-}1$	$2.12\mathrm{E}{-}1$

Table 2: Mean

\pm

std of L2RE for data assimilation and parameter inversion.

L1RE	Data assimilation				Parameter inversion
	Poisson	Heat	Wave	Stokes	PInv	HInv	NSInv	EBInv	WInv
PINN	$1.85\mathrm{E}{-}1$	$5.34\mathrm{E}{-}1$	$1.45\mathrm{E}{-}1$	$6.04\mathrm{E}{-}1$	$6.44\mathrm{E}{-}2$	$1.03\mathrm{E}{-}1$	$3.80\mathrm{E}{+}0$	$2.50\mathrm{E}{-}1$	$9.94\mathrm{E}{-}2$
PINN	$\pm 6.13\mathrm{E}{-}3$	$\pm 2.17\mathrm{E}{-}1$	$\pm 5.50\mathrm{E}{-}2$	$\pm 7.30\mathrm{E}{-}3$	$\pm 1.97\mathrm{E}{-}2$	$\pm 3.06\mathrm{E}{-}3$	$\pm 8.43\mathrm{E}{-}1$	$\pm 1.96\mathrm{E}{-}2$	$\pm 2.60\mathrm{E}{-}3$
P-PINN	$\mathbf{6.04\mathrm{E}{-}2}$	$\mathbf{1.35\mathrm{E}{-}2}$	$\mathbf{2.50\mathrm{E}{-}2}$	$\mathbf{1.17\mathrm{E}{-}1}$	$\mathbf{1.94\mathrm{E}{-}2}$	$\mathbf{6.20\mathrm{E}{-}2}$	$1.49\mathrm{E}{+}0$	$\mathbf{6.95\mathrm{E}{-}2}$	$\mathbf{5.72\mathrm{E}{-}2}$
P-PINN	$\mathbf{\pm 2.23\mathrm{E}{-}2}$	$\mathbf{\pm 6.63\mathrm{E}{-}3}$	$\mathbf{\pm 2.08\mathrm{E}{-}3}$	$\mathbf{\pm 1.49\mathrm{E}{-}2}$	$\mathbf{\pm 2.26\mathrm{E}{-}3}$	$\mathbf{\pm 3.01\mathrm{E}{-}3}$	$\pm 5.30\mathrm{E}{-}1$	$\mathbf{\pm 7.14\mathrm{E}{-}3}$	$\mathbf{\pm 3.46\mathrm{E}{-}3}$
B-PINN	$3.37\mathrm{E}{-}1$	$4.06\mathrm{E}{-}1$	$7.16\mathrm{E}{-1}$	$9.95\mathrm{E}{-}1$	$4.67\mathrm{E}{-}1$	$2.24\mathrm{E}{-}1$	$2.26\mathrm{E}{+}0$	$7.82\mathrm{E}{-}1$	$2.12\mathrm{E}{-}1$

Table 3: Mean

\pm

std of L1RE.

MSE	Data assimilation				Parameter inversion
	Poisson	Heat	Wave	Stokes	PInv	HInv	NSInv	EBInv	WInv
PINN	$6.68\mathrm{E}{-}2$	$2.00\mathrm{E}{-}1$	$7.54\mathrm{E}{-}3$	$2.08\mathrm{E}{-}1$	$6.24\mathrm{E}{-}2$	$7.44\mathrm{E}{-}2$	$2.73\mathrm{E}{-}1$	$6.22\mathrm{E}{-}2$	$3.95\mathrm{E}{-}2$
PINN	$\pm 7.14\mathrm{E}{-}3$	$\pm 1.42\mathrm{E}{-}1$	$\pm 5.87\mathrm{E}{-}3$	$\pm 6.80\mathrm{E}{-}3$	$\pm 1.90\mathrm{E}{-}2$	$\pm 2.62\mathrm{E}{-}3$	$\pm 1.47\mathrm{E}{-}1$	$\pm 9.75\mathrm{E}{-}3$	$\pm 2.03\mathrm{E}{-}3$
P-PINN	$\mathbf{7.50\mathrm{E}{-}3}$	$\mathbf{6.43\mathrm{E}{-}4}$	$\mathbf{1.57\mathrm{E}{-}4}$	$\mathbf{6.60\mathrm{E}{-}3}$	$\mathbf{1.32\mathrm{E}{-}4}$	$\mathbf{2.82\mathrm{E}{-}2}$	$2.48\mathrm{E}{-}2$	$\mathbf{4.84\mathrm{E}{-}3}$	$\mathbf{1.29\mathrm{E}{-}2}$
P-PINN	$\mathbf{\pm 5.73\mathrm{E}{-}3}$	$\mathbf{\pm 9.67\mathrm{E}{-}3}$	$\mathbf{\pm 2.89\mathrm{E}{-}5}$	$\mathbf{\pm 2.10\mathrm{E}{-}3}$	$\mathbf{\pm 3.12\mathrm{E}{-}5}$	$\mathbf{\pm 5.02\mathrm{E}{-}2}$	$\pm 1.16\mathrm{E}{-}2$	$\mathbf{\pm 9.92\mathrm{E}{-}4}$	$\mathbf{\pm 1.58\mathrm{E}{-}3}$
B-PINN	$1.46\mathrm{E}{-}1$	$8.26\mathrm{E}{-}2$	$1.24\mathrm{E}{-}1$	$2.29\mathrm{E}{-}1$	$1.32\mathrm{E}{-}4$	$3.19\mathrm{E}{-}1$	$1.21\mathrm{E}{+}0$	$6.13\mathrm{E}{-}1$	$2.78\mathrm{E}{-}2$

Table 4: Mean

\pm

std of MSE.

Method	$\boldsymbol{\beta_{1}}$ error	$\boldsymbol{\beta_{2}}$ error
PINN	$1.79\mathrm{E}{-}1\;\pm 1.04\mathrm{E}{-}2$	$9.03\mathrm{E}{-}3\;\pm 2.10\mathrm{E}{-}4$
P-PINN	$4.56\mathrm{E}{-}2\;\pm 1.19\mathrm{E}{-}2$	$7.05\mathrm{E}{-}4\;\pm 4.88\mathrm{E}{-}4$
B-PINN	$2.46\mathrm{E}{-}1\;\pm 8.49\mathrm{E}{-}2$	$5.02\mathrm{E}{-}4\;\pm 4.54\mathrm{E}{-}4$
D-PINN	$\mathbf{2.23\mathrm{E}{-}2\;\pm 4.84\mathrm{E}{-}3}$	$\mathbf{1.03\mathrm{E}{-}4\;\pm 2.97\mathrm{E}{-}6}$

Table 5: Mean

\pm

std of MSE of

\beta_{1}

and

\beta_{2}

for NSInv.

Max error	Data assimilation				Parameter inversion
	Poisson	Heat	Wave	Stokes	PInv	HInv	NSInv	EBInv	WInv
PINN	$1.66\mathrm{E}{+}0$	$1.08\mathrm{E}{+}0$	$2.42\mathrm{E}{-}1$	$3.32\mathrm{E}{+}0$	$7.89\mathrm{E}{-}2$	$8.49\mathrm{E}{-}1$	$3.46\mathrm{E}{+}0$	$2.50\mathrm{E}{-}1$	$1.99\mathrm{E}{-}1$
PINN	$\pm 3.01\mathrm{E}{-}1$	$\pm 3.54\mathrm{E}{-}1$	$\pm 1.15\mathrm{E}{-}1$	$\pm 1.52\mathrm{E}{-}2$	$\pm 2.06\mathrm{E}{-}2$	$\pm 6.03\mathrm{E}{-}2$	$\pm 1.03\mathrm{E}{+}0$	$\pm 1.96\mathrm{E}{-}2$	$\pm 5.11\mathrm{E}{-}3$
P-PINN	$\mathbf{4.73\mathrm{E}{-}1}$	$\mathbf{4.28\mathrm{E}{-}2}$	$\mathbf{3.80\mathrm{E}{-}2}$	$\mathbf{6.00\mathrm{E}{-}1}$	$\mathbf{2.97\mathrm{E}{-}2}$	$\mathbf{5.81\mathrm{E}{-}1}$	$\mathbf{5.19\mathrm{E}{-}1}$	$\mathbf{6.95\mathrm{E}{-}2}$	$\mathbf{1.14\mathrm{E}{-}1}$
P-PINN	$\mathbf{\pm 1.32\mathrm{E}{-}1}$	$\mathbf{\pm 3.04\mathrm{E}{-}2}$	$\mathbf{\pm 4.22\mathrm{E}{-}3}$	$\mathbf{\pm 1.28\mathrm{E}{-}1}$	$\mathbf{\pm 5.86\mathrm{E}{-}3}$	$\mathbf{\pm 2.54\mathrm{E}{-}2}$	$\mathbf{\pm 5.69\mathrm{E}{-}2}$	$\mathbf{\pm 7.14\mathrm{E}{-}3}$	$\mathbf{\pm 6.91\mathrm{E}{-}3}$
B-PINN	$1.09\mathrm{E}{+}0$	$4.34\mathrm{E}{-}1$	$7.96\mathrm{E}{-}1$	$2.11\mathrm{E}{+}0$	$5.63\mathrm{E}{-}1$	$4.48\mathrm{E}{-}1$	$3.86\mathrm{E}{+}0$	$7.82\mathrm{E}{-}1$	$4.25\mathrm{E}{-}1$

Table 6: Mean

\pm

std of MAE.

fMSE-L	Data assimilation				Parameter inversion
	Poisson	Heat	Wave	Stokes	PInv	HInv	NSInv	EBInv	WInv
PINN	$2.80\mathrm{E}{+}2$	$5.50\mathrm{E}{+}2$	$1.08\mathrm{E}{+}2$	$8.29\mathrm{E}{+}2$	$3.12\mathrm{E}{+}1$	$1.39\mathrm{E}{+}2$	$1.20\mathrm{E}{+}3$	—	—
PINN	$\pm 2.21\mathrm{E}{+}1$	$\pm 2.16\mathrm{E}{+}2$	$\pm 5.03\mathrm{E}{+}1$	$\pm 1.32\mathrm{E}{+}1$	$\pm 9.64\mathrm{E}{+}0$	$\pm 2.54\mathrm{E}{+}1$	$\pm 7.25\mathrm{E}{+}2$
P-PINN	$\mathbf{1.11\mathrm{E}{+}2}$	$\mathbf{1.66\mathrm{E}{+}1}$	$\mathbf{1.54\mathrm{E}{+}1}$	$\mathbf{1.12\mathrm{E}{+}2}$	$\mathbf{1.04\mathrm{E}{+}1}$	$\mathbf{5.89\mathrm{E}{+}1}$	$4.20\mathrm{E}{+}2$	—	—
P-PINN	$\mathbf{\pm 4.52\mathrm{E}{+}1}$	$\mathbf{\pm 9.11\mathrm{E}{+}0}$	$\mathbf{\pm 7.15\mathrm{E}{-}1}$	$\mathbf{\pm 1.73\mathrm{E}{+}1}$	$\mathbf{\pm 1.34\mathrm{E}{+}0}$	$\mathbf{\pm 1.22\mathrm{E}{+}1}$	$\pm 1.17\mathrm{E}{+}2$
B-PINN	$1.97\mathrm{E}{+}3$	$3.88\mathrm{E}{+}2$	$6.96\mathrm{E}{+}2$	$2.59\mathrm{E}{+}3$	$4.92\mathrm{E}{+}4$	$2.11\mathrm{E}{+}2$	$2.81\mathrm{E}{+}3$	—	—

Table 7: Mean

\pm

std of fMSE-L.

fMSE-M	Data assimilation				Parameter inversion
	Poisson	Heat	Wave	Stokes	PInv	HInv	NSInv	EBInv	WInv
PINN	$7.45\mathrm{E}{+}1$	$5.12\mathrm{E}{+}1$	$6.77\mathrm{E}{+}0$	$9.26\mathrm{E}{+}1$	$5.60\mathrm{E}{-}1$	$3.94\mathrm{E}{+}0$	$4.87\mathrm{E}{+}1$	—	—
PINN	$\pm 5.82\mathrm{E}{+}0$	$\pm 2.09\mathrm{E}{+}1$	$\pm 2.54\mathrm{E}{+}0$	$\pm 4.38\mathrm{E}{-}1$	$\pm 1.76\mathrm{E}{-}1$	$\pm 5.12\mathrm{E}{-}1$	$\pm 5.16\mathrm{E}{+}0$
P-PINN	$\mathbf{1.16\mathrm{E}{+}1}$	$\mathbf{7.40\mathrm{E}{-}1}$	$\mathbf{1.91\mathrm{E}{+}0}$	$\mathbf{1.19\mathrm{E}{+}1}$	$\mathbf{1.90\mathrm{E}{-}1}$	$\mathbf{2.18\mathrm{E}{+}0}$	$\mathbf{5.59\mathrm{E}{+}0}$	—	—
P-PINN	$\mathbf{\pm 6.25\mathrm{E}{+}0}$	$\mathbf{\pm 4.54\mathrm{E}{-}1}$	$\mathbf{\pm 1.82\mathrm{E}{-}1}$	$\mathbf{\pm 2.25\mathrm{E}{+}0}$	$\mathbf{\pm 5.87\mathrm{E}{-}2}$	$\mathbf{\pm 5.48\mathrm{E}{-}2}$	$\mathbf{\pm 9.33\mathrm{E}{-}1}$
B-PINN	$1.85\mathrm{E}{+}2$	$8.53\mathrm{E}{+}0$	$5.42\mathrm{E}{+}0$	$1.74\mathrm{E}{+}2$	$2.52\mathrm{E}{+}2$	$1.55\mathrm{E}{+}1$	$1.21\mathrm{E}{+}2$	—	—

Table 8: Mean

\pm

std of fMSE-M.

fMSE-H	Data assimilation				Parameter inversion
	Poisson	Heat	Wave	Stokes	PInv	HInv	NSInv	EBInv	WInv
PINN	$7.47\mathrm{E}{+}0$	$6.06\mathrm{E}{+}0$	$7.15\mathrm{E}{-}1$	$1.33\mathrm{E}{+}1$	$8.35\mathrm{E}{-}2$	$5.41\mathrm{E}{-}1$	$2.13\mathrm{E}{+}0$	—	—
PINN	$\pm 3.85\mathrm{E}{-}1$	$\pm 2.92\mathrm{E}{+}0$	$\pm 3.23\mathrm{E}{-}1$	$\pm 9.74\mathrm{E}{-}2$	$\pm 2.25\mathrm{E}{-}2$	$\pm 7.56\mathrm{E}{-}2$	$\pm 1.94\mathrm{E}{-}1$
P-PINN	$\mathbf{1.57\mathrm{E}{+}0}$	$\mathbf{9.75\mathrm{E}{-}2}$	$\mathbf{1.76\mathrm{E}{-}1}$	$\mathbf{1.72\mathrm{E}{+}0}$	$\mathbf{1.28\mathrm{E}{-}2}$	$\mathbf{3.46\mathrm{E}{-}1}$	$\mathbf{6.51\mathrm{E}{-}1}$	—	—
P-PINN	$\mathbf{\pm 8.52\mathrm{E}{-}1}$	$\mathbf{\pm 5.64\mathrm{E}{-}2}$	$\mathbf{\pm 2.05\mathrm{E}{-}2}$	$\mathbf{\pm 3.06\mathrm{E}{-}1}$	$\mathbf{\pm 9.72\mathrm{E}{-}4}$	$\mathbf{\pm 5.04\mathrm{E}{-}2}$	$\mathbf{\pm 1.47\mathrm{E}{-}1}$
B-PINN	$1.35\mathrm{E}{+}1$	$4.26\mathrm{E}{-}1$	$2.70\mathrm{E}{-}1$	$1.42\mathrm{E}{+}1$	$7.84\mathrm{E}{+}0$	$3.66\mathrm{E}{+}0$	$1.07\mathrm{E}{+}0$	—	—

Table 9: Mean

\pm

std of fMSE-H.

Panel of Figure 3 — Figure 3: Solution fields for the one-dimensional wave data assimilation problem: (a) standard PINN prediction, which roughly identifies the correct locations of the wave lobes but is contaminated by pronounced spurious oscillations; (b) exact solution $u(x,t)$ ; (c) P-PINN prediction after selective pruning and fine-tuning, which closely matches the smooth standing-wave structure of the exact solution.

Panel of Figure 4 — Figure 4: Heat equation data assimilation: (a) standard PINN prediction; (b) exact solution $u(x,t)$ ; (c) P-PINN prediction after selective pruning and fine-tuning.

Panel of Figure 5 — Figure 5: Poisson data assimilation: (a) standard PINN prediction; (b) exact solution $u(x,y)$ ; (c) P-PINN prediction.

Panel of Figure 6 — Figure 6: Stokes data assimilation: velocity and pressure components. Rows from top to bottom: $u$ -component, $v$ -component, and pressure $p$ . Columns from left to right: standard PINN prediction, exact solution, and P-PINN prediction.

Panel of Figure 7 — Figure 7: Diffusion coefficient $a(x,y)$ for the Poisson inverse problem: (a) standard PINN prediction; (b) ground truth; (c) P-PINN prediction.

Panel of Figure 8 — Figure 8: Diffusion coefficient $a(x,y)$ for the heat inverse problem: (a) standard PINN prediction; (b) ground truth; (c) P-PINN prediction.

(a) Time (s)
Problem	RT	FT	P-PINN
Heat	$7.24\mathrm{E}{+}1$	$1.12\mathrm{E}{+}1$	$1.62\mathrm{E}{+}1$
Wave	$9.98\mathrm{E}{+}1$	$1.97\mathrm{E}{+}1$	$2.00\mathrm{E}{+}1$
PInv	$1.65\mathrm{E}{+}2$	$1.69\mathrm{E}{+}1$	$1.70\mathrm{E}{+}1$
NSInv	$3.22\mathrm{E}{+}2$	$3.03\mathrm{E}{+}1$	$3.18\mathrm{E}{+}1$

Table 10: Training time and test L2RE for three strategies across four PDE problems.

Strategy	Heat	Wave	PInv	NSInv
Iterative	$\mathbf{3.26\mathrm{E}{-}2}$	$\mathbf{8.88\mathrm{E}{-}2}$	$\mathbf{1.82\mathrm{E}{-}2}$	$\mathbf{6.44\mathrm{E}{-}1}$
	$\pm 1.56\mathrm{E}{-}4$	$\pm 4.38\mathrm{E}{-}4$	$\pm 2.17\mathrm{E}{-}3$	$\pm 2.67\mathrm{E}{-}1$
Single	$7.28\mathrm{E}{-}1$	$\mathbf{8.81\mathrm{E}{-}2}$	$1.94\mathrm{E}{-}2$	$8.87\mathrm{E}{-}1$
	$\pm 1.77\mathrm{E}{-}2$	$\pm 6.05\mathrm{E}{-}4$	$\pm 4.67\mathrm{E}{-}3$	$\pm 1.32\mathrm{E}{-}1$

Table 11: Effect of pruning strategies on L2RE (mean

\pm

std) across four PDEs.

Criterion	Heat	Wave	PInv	NSInv
Bias	$3.26\mathrm{E}{-}2$	$\mathbf{8.88\mathrm{E}{-}2}$	$\mathbf{1.82\mathrm{E}{-}2}$	$\mathbf{6.44\mathrm{E}{-}1}$
	$\pm 1.56\mathrm{E}{-}4$	$\pm 4.38\mathrm{E}{-}4$	$\pm 2.17\mathrm{E}{-}3$	$\pm 2.67\mathrm{E}{-}1$
Abs	$2.59\mathrm{E}{-}2$	$1.01\mathrm{E}{-}1$	$3.52\mathrm{E}{-}2$	$2.01\mathrm{E}{+}0$
	$\pm 1.52\mathrm{E}{-}4$	$\pm 7.77\mathrm{E}{-}4$	$\pm 3.14\mathrm{E}{-}3$	$\pm 6.40\mathrm{E}{-}1$
RMS	$\mathbf{2.19\mathrm{E}{-}2}$	$8.21\mathrm{E}{-}1$	$3.27\mathrm{E}{-}2$	$1.48\mathrm{E}{+}0$
	$\pm 3.80\mathrm{E}{-}5$	$\pm 3.14\mathrm{E}{-}3$	$\pm 9.36\mathrm{E}{-}3$	$\pm 5.47\mathrm{E}{-}2$

Table 12: Effect of neuron-importance criteria on L2RE (mean

\pm

std) across four PDEs.

Noise std	Heat		Wave
	PINN	P-PINN	PINN	P-PINN
0.25	$5.25\mathrm{E}{-}1$	$3.71\mathrm{E}{-}2$	$4.96\mathrm{E}{-}2$	$9.90\mathrm{E}{-}3$
0.25	$\pm 1.99\mathrm{E}{-}2$	$\pm 9.20\mathrm{E}{-}3$	$\pm 1.42\mathrm{E}{-}2$	$\pm 4.20\mathrm{E}{-}3$
0.50	$1.06\mathrm{E}{+}0$	$4.89\mathrm{E}{-}2$	$1.94\mathrm{E}{-}1$	$2.89\mathrm{E}{-}2$
0.50	$\pm 1.02\mathrm{E}{+}0$	$\pm 8.80\mathrm{E}{-}3$	$\pm 1.29\mathrm{E}{-}1$	$\pm 1.09\mathrm{E}{-}2$
0.75	$1.64\mathrm{E}{+}0$	$1.77\mathrm{E}{-}1$	$1.30\mathrm{E}{-}1$	$2.23\mathrm{E}{-}2$

Table 13: Mean

\pm

std of L2RE under varying noise levels for heat and wave data assimilation.

Figure 9: L2RE for four inverse problems (NSInv, PInv, Heat, Wave) as a function of (left) the number of pruned layers and (right) the percentage of retained data (with five layers pruned). The label “base” denotes the baseline PINN trained on the full noisy dataset. Solid lines show means over three runs; shaded bands indicate $\pm 1$ standard deviation. Hollow markers in the right panel highlight the retention ratios used in the main experiments.

Physics-Informed ML

Physics-informed machine learning combines data-driven models with physical laws to solve, predict, and infer scientific systems.

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Apr 23, 2026

Transferable Physics-Informed Representations via Closed-Form Head Adaptation

Physics-informed neural networks (PINNs) have garnered significant interest for their potential in solving partial differential equations (PDEs) that govern a wide range of physical phenomena. By incorporating physical laws into the learning process, PINN models have demonstrated the ability to learn physical outcomes reasonably well. However, current PINN approaches struggle to predict or solve new PDEs effectively when there is a lack of training examples, indicating they do not generalize well to unseen problem instances. In this paper, we present a transferable learning approach for PINNs premised on a fast Pseudoinverse PINN framework (Pi-PINN). Pi-PINN learns a transferable physics-informed representation in a shared embedding space and enables rapid solving of both known and unknown PDE instances via closed-form head adaptation using a least-squares-optimal pseudoinverse under PDE constraints. We further investigate the synergies between data-driven multi-task learning loss and physics-informed loss, providing insights into the design of more performant PINNs. We demonstrate the effectiveness of Pi-PINN on various PDE problems, including Poisson's equation, Helmholtz equation, and Burgers' equation, achieving fast and accurate physics-informed solutions without requiring any data for unseen instances. Pi-PINN can produce predictions 100-1000 times faster than a typical PINN, while producing predictions with 10-100 times lower relative error than a typical data-driven model even with only two training samples. Overall, our findings highlight the potential of transferable representations with closed-form head adaptation to enhance the efficiency and generalization of PINNs across PDE families and scientific and engineering applications.

Jian Cheng Wong, Isaac Yin Chung Lai, Pao-Hsiung Chiu +3

Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore · National University of Singapore (NUS), Singapore · Indian Institute of Technology Goa (IIT Goa), India +1

Representation Learning PDE Solving Physics-Informed ML

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Feb 23, 2026

Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correction

Physics-informed neural networks (PINNs) have emerged as a promising mesh-free paradigm for solving partial differential equations, yet adoption in science and engineering is limited by slow training and modest accuracy relative to modern numerical solvers. We introduce the Sequential Correction Algorithm for Learning Efficient PINN (Scale-PINN), a learning strategy that bridges modern physics-informed learning with numerical algorithms. Scale-PINN incorporates the iterative residual-correction principle, a cornerstone of numerical solvers, directly into the loss formulation, marking a paradigm shift in how PINN losses can be conceived and constructed. This integration enables Scale-PINN to achieve unprecedented convergence speed across PDE problems from different physics domain, including reducing training time on a challenging fluid-dynamics problem for state-of-the-art PINN from hours to sub-2 minutes while maintaining superior accuracy, and enabling application to representative problems in aerodynamics and urban science. By uniting the rigor of numerical methods with the flexibility of deep learning, Scale-PINN marks a significant leap toward the practical adoption of PINNs in science and engineering through scalable, physics-informed learning. Codes are available at https://github.com/chiuph/SCALE-PINN.

Pao-Hsiung Chiu, Jian Cheng Wong, Chin Chun Ooi +3

Institute of High Performance Computing, A*STAR, Singapore. · Centre for Frontier AI Research, A*STAR, Singapore. · School of Mechanical Engineering, Tianjin University, China. +1

Neural PDE Solvers Physics-Informed ML Neural Network Optimization

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Aug 25, 2026

Physics-Informed Error Field Learning: A Post-Training Optimization Framework for Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have emerged as an important class of numerical methods for solving partial differential equations (PDEs). However, during the late-stage optimization process, further parameter updates often yield diminishing accuracy improvements while increasing computational costs. To address this issue, this paper proposes a Physics-Informed Error Field Learning (PIEFL) framework for PINNs. Unlike conventional approaches that continuously approximate the solution field using a single network, PIEFL introduces an auxiliary error network after the primary network achieves satisfactory accuracy and shifts the learning objective from the solution field to the error field. By deriving error control equations under physical constraints, the error network learns the discrepancy between the current approximation and the exact solution, and the learned error correction is combined with the primary prediction to improve solution accuracy. The proposed framework avoids continuous optimization of the entire solution space and focuses computational resources on correcting existing prediction errors. Moreover, PIEFL requires no modification to the primary network architecture, making it compatible with existing PINN models and applicable as a general post-training optimization strategy. Numerical experiments on representative PDEs demonstrate that PIEFL achieves higher solution accuracy under the same computational budget, validating its effectiveness in improving the performance of PINNs.

Jiuyun Sun, Yong Zhang

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qianwangang Road 579, Qingdao, 266590, Shandong, China

Neural PDE Solvers Physics-Informed ML Neural Network Optimization

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Abstract

Physics-Informed ML

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Transferable Physics-Informed Representations via Closed-Form Head Adaptation

Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correction

Physics-Informed Error Field Learning: A Post-Training Optimization Framework for Physics-Informed Neural Networks