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arxiv: 2605.30910 · v1 · pith:4JEXWX2Knew · submitted 2026-05-29 · 💻 cs.LG

PINNs Failure Modes are Overfitting

Nigel T. Andersen , Takashi Matsubara This is my paper

Pith reviewed 2026-06-28 23:53 UTC · model grok-4.3

classification 💻 cs.LG
keywords physics-informed neural networksPINNsoverfittingfailure modesresidual visualizationregularizationdouble backpropagationPDE solvers
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The pith

Failure modes in Physics-Informed Neural Networks result from overfitting to collocation points.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Physics-Informed Neural Networks minimize a residual loss to solve PDEs but sometimes converge to wrong solutions despite low loss. Direct visualization of the residual across the domain shows low values only at the discrete collocation points, with high residuals elsewhere. This localized minimization indicates overfitting. Applying regularization causes the failure modes to disappear. Extending double backpropagation to the full set of residuals yields state-of-the-art performance on standard failure cases with up to 23 times fewer collocation points.

Core claim

By directly visualizing the residual, failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Extending double backpropagation over the full set of residuals achieves state-of-the-art performance on four standard failure mode equations with up to 23× fewer collocation points and a vanilla architecture.

What carries the argument

Visualization of the residual field over the full domain, which reveals minimization only at collocation points.

Load-bearing premise

The mismatch between low residual at collocation points and high residual elsewhere is the direct cause of convergence to incorrect solutions.

What would settle it

Train a PINN on a known failure-mode equation, apply regularization, and check whether the residual becomes uniformly low everywhere and the solution matches the correct one; if an incorrect solution persists despite uniform low residual, the overfitting claim is challenged.

Figures

Figures reproduced from arXiv: 2605.30910 by Nigel T. Andersen, Takashi Matsubara.

Figure 1
Figure 1. Figure 1: Exact solution to the wave equaiton (a), compared to the PINNs failure mode (b). Un [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Examples of failures modes at (a) FP32 (convection equation), and (b) FP64 (wave equation). [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Failure and success for the convection equation at FP32. Exact solution is shown in (a), [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the solution to the Allen-Cahn equation (a), with PINNs (b)-(d). Some [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Regularization removes PINNs failure modes, while FP64 improves optimization. Com [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The unregularized case shows failure caused by overfitting. All three of the regularization [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of vanilla PINNs on the β = 50 convection equation at FP32. (a) unregularized PINNs fails due to overfitting of the residual. (b) Double PINNs succeed after a little over 5000 iterations of L-BFGS optimization. (c) L 2 regularized PINNs also succeeds, but requires more iterations, and maintains a higher test loss. (d) L 1 PINNs avoid failure from overfitting, but are difficult to optimize, resul… view at source ↗
read the original abstract

Physics-Informed Neural Networks (PINNs) are a common class of machine learning-based partial differential equation (PDE) solvers which train a network to represent a solution by minimizing a residual loss that encodes the PDE. Despite their successes, they are known to fail on certain simple equations, converging to an incorrect solution despite low loss. These failure modes have garnered significant attention in the literature over the past several years, motivating both architectural and optimization based solutions. By directly visualizing the residual, we show that failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Finally, we extend double backpropagation over the full set of residuals, and use it to achieve state-of-the-art performance on four standard failure mode equations with up to $23\times$ fewer collocation points and a vanilla architecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that PINN failure modes on simple PDEs—convergence to incorrect solutions despite low training loss—are caused by overfitting, where the PDE residual is minimized only at collocation points but remains high elsewhere. This is supported by direct residual visualizations; applying regularization eliminates the mismatch and the failures. The authors further extend double backpropagation to the full residual set and report state-of-the-art results on four standard benchmark equations, using up to 23× fewer collocation points with a vanilla network architecture.

Significance. If the interpretation is substantiated, the work supplies a concrete mechanistic account of a well-known PINN pathology together with a lightweight regularization fix and an efficient double-backprop variant. The reported gains in accuracy and data efficiency on established test cases would be of direct practical value to the PINN literature.

major comments (2)
  1. [Abstract and visualization/results sections] The central claim equates the observed low residual on collocation points and high residual elsewhere with 'overfitting' that directly produces incorrect convergence. Visualization and the fact that regularization removes both the mismatch and the failure are presented as evidence, yet no ablation isolates this mechanism from alternatives (e.g., optimizer reaching a poor local minimum due to loss-landscape stiffness or gradient pathologies). This causal link is load-bearing for the title and abstract conclusions.
  2. [Method / experimental results on the four benchmark equations] The double-backpropagation extension is reported to achieve SOTA performance, but the manuscript does not detail how the 'full set of residuals' is constructed, which terms receive the second derivative, or how this differs quantitatively from prior double-backprop uses in PINNs. Without these specifics the performance claims cannot be fully evaluated.
minor comments (2)
  1. [Figures] Figure captions and axis labels for the residual visualizations should explicitly state the PDE, collocation-point density, and network architecture used in each panel to allow direct replication.
  2. [Abstract and results tables] The abstract states 'up to 23× fewer collocation points'; the corresponding table or text should report the exact counts and the baseline methods against which the factor is measured.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and visualization/results sections] The central claim equates the observed low residual on collocation points and high residual elsewhere with 'overfitting' that directly produces incorrect convergence. Visualization and the fact that regularization removes both the mismatch and the failure are presented as evidence, yet no ablation isolates this mechanism from alternatives (e.g., optimizer reaching a poor local minimum due to loss-landscape stiffness or gradient pathologies). This causal link is load-bearing for the title and abstract conclusions.

    Authors: The direct visualization of the PDE residual across the entire domain explicitly demonstrates that the loss is minimized exclusively at collocation points while remaining high elsewhere; this constitutes overfitting by definition in the context of the residual loss. Regularization eliminates both the spatial mismatch and the incorrect convergence, providing supporting evidence for causality. While alternative mechanisms such as loss-landscape stiffness cannot be ruled out in every possible scenario, the presented visualizations and regularization results isolate the overfitting effect as the operative mechanism here. We will add a dedicated discussion paragraph addressing potential alternative explanations in the revised manuscript. revision: partial

  2. Referee: [Method / experimental results on the four benchmark equations] The double-backpropagation extension is reported to achieve SOTA performance, but the manuscript does not detail how the 'full set of residuals' is constructed, which terms receive the second derivative, or how this differs quantitatively from prior double-backprop uses in PINNs. Without these specifics the performance claims cannot be fully evaluated.

    Authors: We agree that the current manuscript lacks sufficient implementation details. In the revised version we will expand the Methods section to specify (i) how the full residual set is assembled from the PDE operator, (ii) precisely which residual terms receive the second-derivative computation, and (iii) a quantitative comparison (in terms of additional gradient operations and memory) to earlier double-backpropagation usages in the PINN literature. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on direct residual visualization and regularization experiments

full rationale

The paper's central argument equates observed low residual at collocation points (high elsewhere) with overfitting as the cause of failure modes, supported by visualizations and the empirical effect of regularization (plus an extension of double backpropagation). No derivation reduces a claimed prediction or result to its own fitted inputs by construction, no self-citation is invoked as a load-bearing uniqueness theorem, and no ansatz or renaming is smuggled in. The derivation chain is observational and experimental rather than self-referential, making the result self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities used in the work.

pith-pipeline@v0.9.1-grok · 5671 in / 1033 out tokens · 35374 ms · 2026-06-28T23:53:41.128687+00:00 · methodology

discussion (0)

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