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arxiv: 2606.06164 · v1 · pith:WUS7VYK3new · submitted 2026-06-04 · 💻 cs.LG · physics.comp-ph

On the training of physics-informed neural operators for solving parametric partial differential equations

Nanxi Chen , Chuanjie Cui , Airong Chen , Sifan Wang , Rujin Ma This is my paper

Pith reviewed 2026-06-28 03:13 UTC · model grok-4.3

classification 💻 cs.LG physics.comp-ph
keywords physics-informed neural operatorsparametric partial differential equationsneural operatorstraining pipelineoptimization challengesCViTDeepONetFNO
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The pith

A carefully designed physics-informed training pipeline for neural operators can match or outperform purely data-driven methods on parametric PDEs.

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

The paper tests key elements of PINO training such as architecture, optimizer, loss balancing, and point sampling across DeepONet, FNO, and CViT on five parametric PDE systems. It shows that optimization problems like gradient conflicts carry over from PINN training but respond to the same fixes. With these adjustments, physics-informed supervision alone reaches or exceeds the accuracy of training that requires paired simulation data.

Core claim

Across the tested backbones and PDE families, CViT delivers the most stable performance; standard mitigation methods for gradient conflicts and causal violations transfer directly to the operator setting; and a tuned physics-informed objective can equal or surpass data-driven neural operator accuracy under multiple data regimes.

What carries the argument

The physics-informed training pipeline that replaces paired input-output data with governing PDE residuals as the supervision signal, applied to neural operator architectures for learning parametric solution maps.

If this is right

  • CViT provides consistently strong and stable performance across the considered benchmarks.
  • Mitigation algorithms developed for PINNs remain effective when training PINOs.
  • Physics-informed training can match and in some cases outperform purely data-driven neural operators under different data regimes.
  • The empirical findings supply a practical pipeline for efficient and robust physics-informed operator learning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same pipeline might reduce the need for expensive high-fidelity simulation data when learning operators for more complex or higher-dimensional PDEs.
  • The transfer of mitigation techniques suggests the optimization challenges are largely architecture-independent within the neural-operator family.
  • Future tests could check whether these patterns persist when the number of training instances or the model size is scaled significantly beyond the reported experiments.

Load-bearing premise

The five parametric PDE systems and benchmarks chosen are representative enough that the observed performance patterns and mitigation effectiveness will hold for other parametric PDE problems of practical interest.

What would settle it

On a new parametric PDE system outside the original five, applying the recommended pipeline and mitigations still yields lower accuracy than a data-driven baseline trained on the same number of simulation pairs.

Figures

Figures reproduced from arXiv: 2606.06164 by Airong Chen, Chuanjie Cui, Nanxi Chen, Rujin Ma, Sifan Wang.

Figure 1
Figure 1. Figure 1: Overview of the training pipeline for physics-informed neural operators. The input function is encoded by a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic comparison of the three neural operator architectures studied in this work. DeepONet combines [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: L 2 norms of the gradients of the four loss components (two PDE residual terms: momentum and continuity; two initial condition terms: height and velocity) during training on the shallow water benchmark with PI-DeepONet. Left: Unweighted losses. The momentum residual gradient dominates, exceeding the IC velocity gradient by more than two orders of magnitude throughout training. Right: GradNorm-weighted loss… view at source ↗
Figure 4
Figure 4. Figure 4: PDE residual loss per time segment at training steps 10k, 20k, and 40k for PI-CViT on the wave equation. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Gradient alignment score [42] between different loss components during training on the shallow water benchmark with PI-DeepONet. Adam’s alignment collapses after approximately 5,000 steps and remains near zero, indicating severe directional gradient conflicts on this multi-objective physics-informed loss. SOAP, by operating in a preconditioned eigenbasis, maintains a consistently high alignment score of ap… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the four training regimes defined by the use of supervised data and by where the physics [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative L 2 error distributions across all benchmark PDEs. Each violin shows the full distribution over 100 test samples. PI-CViT consistently achieves lower errors and tighter distributions compared to PI-DeepONet and PI-FNO. solution at t = 0, weakening the causal structure of the training dynamics, and the network is prone to collapsing to the trivial zero solution [71]. To mitigate this, we employ a w… view at source ↗
Figure 8
Figure 8. Figure 8: Burgers’ equation. The first velocity component [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Wave equation. Scalar solution field v predicted by different physics-informed neural operators compared with the reference solution. the large Coriolis parameter, which drive rapid geostrophic adjustment and fast multi-scale oscillations throughout the time window [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Shallow water equations. Free-surface height [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Ice melting problem. Phase-field variable [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Lid-driven cavity flow. Streamlines and velocity magnitude predicted by different physics-informed neural [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Ablation study on optimizer selection. Test error convergence curves during training for SOAP and Adam [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Ablation study on labeled training data. Relative [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Ablation study on weighting schemes. Test error convergence curves during training for PI-CViT with and [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Ablation study on derivative computation method. Test error convergence curves during training for PI-CViT [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FD versus AD on the ice melting benchmark. [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Ablation study on time FiLM conditioning. Test error convergence curves during training for PI-CViT with [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
read the original abstract

Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning. Despite this promise, how to train PINOs efficiently and robustly remains less well-understood than the training of either data-driven neural operators or physics-informed neural networks (PINNs). To bridge this gap, we examine key components of the PINO training pipeline, including architecture design, optimizer choice, loss balancing, and collocation-point sampling strategy. We study three representative operator backbones, Deep Operator Network (DeepONet), Fourier Neural Operator (FNO), and Continuous Vision Transformer (CViT), across five diverse parametric PDE systems. Our results show that CViT provides consistently strong and stable performance across the considered benchmarks. Beyond architecture, we find that several optimization pathologies previously identified in PINN training naturally arise in PINOs, including gradient conflicts and causal violation. We also find that mitigation algorithms developed for PINNs remain effective in the PINO setting. We further compare physics-informed and data-driven training under different data regimes, revealing that a carefully designed physics-informed training pipeline can match, and in some cases, outperform purely data-driven neural operators. Taken together, these findings provide a systematic empirical understanding of the optimization challenges in PINO training and inform a practical pipeline for efficient and robust physics-informed operator learning. Code and data are available at https://github.com/NanxiiChen/PI-CViT.

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

1 major / 1 minor

Summary. The paper examines key components of the training pipeline for physics-informed neural operators (PINOs), including architecture design, optimizer choice, loss balancing, and collocation-point sampling. It evaluates three operator backbones—DeepONet, FNO, and CViT—across five parametric PDE systems, showing that CViT performs strongly, that PINN-like optimization issues arise and can be mitigated, and that a well-designed physics-informed approach can match or outperform data-driven training in some data regimes.

Significance. If the results hold, this provides a systematic empirical guide to training PINOs, highlighting effective practices and the potential for physics-informed methods to reduce reliance on simulation data. The release of code and data supports reproducibility and allows verification of the empirical claims.

major comments (1)
  1. [Abstract and experimental evaluation] The central claim that the physics-informed pipeline can match or outperform data-driven neural operators in general relies on the five chosen PDE systems being representative of the optimization challenges and generalization patterns in other parametric PDEs. The manuscript describes the systems as 'diverse' but provides no explicit argument or additional validation for why the observed performance patterns and mitigation effectiveness would transfer to other problems of practical interest. This is load-bearing for the broader conclusions.
minor comments (1)
  1. [Abstract] The abstract states that 'several optimization pathologies previously identified in PINN training naturally arise in PINOs'; it would be helpful to briefly name the specific pathologies (e.g., gradient conflicts, causal violation) already in the abstract for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and positive recommendation. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and experimental evaluation] The central claim that the physics-informed pipeline can match or outperform data-driven neural operators in general relies on the five chosen PDE systems being representative of the optimization challenges and generalization patterns in other parametric PDEs. The manuscript describes the systems as 'diverse' but provides no explicit argument or additional validation for why the observed performance patterns and mitigation effectiveness would transfer to other problems of practical interest. This is load-bearing for the broader conclusions.

    Authors: We agree that the broader conclusions depend on the representativeness of the five PDEs and that the manuscript provides only a high-level description of them as 'diverse' without an explicit justification or discussion of transferability. The systems were chosen to cover linear/nonlinear, time-dependent/steady-state, 1D/2D, and varying parameter dimensions, but we did not articulate selection criteria or limitations. In revision we will add a concise paragraph (likely in Section 3 or the experimental setup) that (i) states the selection criteria, (ii) notes the common optimization and generalization challenges they instantiate, and (iii) acknowledges that extrapolation to other PDE families remains an open question. This addresses the load-bearing concern without altering the empirical claims. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical benchmark study

full rationale

The paper reports direct experimental comparisons of training pipelines for PINOs (DeepONet, FNO, CViT backbones) across five parametric PDE systems. No derivation, fitted parameter, or prediction is presented; all claims rest on observed performance metrics from explicit benchmarks. No self-definitional equations, no fitted inputs relabeled as predictions, and no load-bearing self-citations that reduce the central empirical claim to prior author work. The analysis is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Empirical benchmarking study; central claims rest on experimental outcomes rather than mathematical derivations or new postulated entities.

pith-pipeline@v0.9.1-grok · 5842 in / 994 out tokens · 53989 ms · 2026-06-28T03:13:29.744608+00:00 · methodology

discussion (0)

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