Hard-constrained Physics-informed Neural Networks for Interface Problems

Michael S. Penwarden; Pratanu Roy; Seung Whan Chung; Stephen T. Castonguay; Sumanta Roy; Yucheng Fu

arxiv: 2604.08453 · v2 · pith:DVEKH3HKnew · submitted 2026-04-09 · 🧮 math.NA · cs.NA· physics.comp-ph

Hard-constrained Physics-informed Neural Networks for Interface Problems

Seung Whan Chung , Stephen T. Castonguay , Sumanta Roy , Michael S. Penwarden , Yucheng Fu , Pratanu Roy This is my paper

Pith reviewed 2026-05-21 09:18 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph

keywords physics-informed neural networksinterface problemshard constraintselliptic equationsnumerical PDEswindowing methodbuffer correction

0 comments

The pith

Hard-constrained PINNs embed interface continuity and flux balance directly into the neural network solution, decoupling enforcement from PDE minimization and removing loss-weight tuning.

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

Physics-informed neural networks struggle with interface problems when continuity and flux conditions are imposed only through soft penalty terms in the loss function, which often leads to imperfect enforcement and requires extensive tuning of loss weights. This paper develops two ansatz-based hard-constrained formulations that build those interface conditions into the structure of the trial solution itself. The windowing approach uses compactly supported subnetworks so the conditions hold automatically, while the buffer approach adds lightweight auxiliary corrections at discrete points. Numerical tests on one- and two-dimensional elliptic interface problems show that these methods deliver higher accuracy at the interfaces and remove the tuning step. Anyone solving PDEs with material discontinuities or sharp transitions would benefit from the simpler training and better fidelity near interfaces.

Core claim

The paper introduces two hard-constrained PINN formulations for interface problems. The windowing approach constructs the trial space from compactly supported windowed subnetworks so that interface continuity and flux balance are satisfied by design. The buffer approach augments unrestricted subnetworks with auxiliary buffer functions that enforce boundary and interface constraints at discrete points through a lightweight correction. Both decouple interface enforcement from PDE residual minimization and yield higher interface fidelity on elliptic benchmarks without loss-weight tuning.

What carries the argument

Ansatz-based hard-constrained formulations (windowing with compactly supported subnetworks and buffer with discrete auxiliary corrections) that embed interface continuity and flux conditions into the solution representation by construction.

If this is right

In one dimension the windowing method reaches errors as low as O(10^{-9}) on simple structured cases.
The buffer method maintains roughly O(10^{-5}) accuracy across varied source terms and interface configurations.
In two dimensions the buffer formulation remains robust while windowing becomes sensitive to overlap and corner effects.
Both formulations remove the need to tune loss weights that soft-constrained PINNs require.

Where Pith is reading between the lines

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

The same hard-constraint construction could be adapted to time-dependent or nonlinear interface problems by updating the window or buffer functions dynamically.
The discrete buffer correction may extend more readily to irregular or high-dimensional geometries than the continuous windowing construction.
Combining the buffer method with existing adaptive sampling techniques in PINNs could further reduce training cost on complex interfaces.

Load-bearing premise

The interface geometry and location must be known exactly in advance so that the window functions or buffer correction points can be defined without adding new approximation error.

What would settle it

Train both the windowing and buffer hard-constrained PINNs on the same one- and two-dimensional elliptic interface benchmark problems used in the paper and check whether the measured interface error norms are lower than those obtained from standard soft-constrained PINNs when no loss-weight tuning is performed.

Figures

Figures reproduced from arXiv: 2604.08453 by Michael S. Penwarden, Pratanu Roy, Seung Whan Chung, Stephen T. Castonguay, Sumanta Roy, Yucheng Fu.

**Figure 2.** Figure 2: Model problem with a slanted interface in a two-dimensional domain: (a) solu [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** Figure 3: The training performance of the windowing approach on the problem in Sec [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

**Figure 4.** Figure 4: The training performance of the buffer approach on the problem in Section 3.1: [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

**Figure 5.** Figure 5: Trained solution of window approach (10) for Problem 4: (a) trained solution; [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗

**Figure 6.** Figure 6: Trained solution of buffer approach (Eq. 38) for Problem 4: (a) trained solution; [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗

**Figure 7.** Figure 7: Boundary enforcement of the buffer approach: (a) the Neumann condition on the [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗

read the original abstract

Physics-informed neural networks (PINNs) have emerged as a flexible framework for solving partial differential equations, but their performance on interface problems remains challenging because continuity and flux conditions are typically imposed through soft penalty terms. The standard soft-constraint formulation leads to imperfect interface enforcement and degraded accuracy near interfaces. We introduce two ansatz-based hard-constrained PINN formulations for interface problems that embed the interface physics into the solution representation and thereby decouple interface enforcement from PDE residual minimization. The first, termed the windowing approach, constructs the trial space from compactly supported windowed subnetworks so that interface continuity and flux balance are satisfied by design. The second, called the buffer approach, augments unrestricted subnetworks with auxiliary buffer functions that enforce boundary and interface constraints at discrete points through a lightweight correction. We study these formulations on one- and two-dimensional elliptic interface benchmarks and compare them with soft-constrained baselines. In one-dimensional problems, hard constraints consistently improve interface fidelity and remove the need for loss-weight tuning; the windowing approach attains very high accuracy (as low as $O(10^{-9})$) on simple structured cases, whereas the buffer approach remains accurate ($\sim O(10^{-5})$) across a wider range of source terms and interface configurations. In two dimensions, the buffer formulation is shown to be more robust because it enforces constraints through a discrete buffer correction, as the windowing construction becomes more sensitive to overlap and corner effects and over-constrains the problem. This positions the buffer method as a straightforward and geometrically flexible approach to complex interface problems.

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.

Desk Editor's Note private letter to a colleague

Hard-constrained windowing and buffer PINN formulations improve 1D interface accuracy and remove loss-weight tuning, but windowing looks fragile in 2D due to geometry sensitivity.

read the letter

This paper's main contribution is a pair of hard-constrained formulations for PINNs that handle interface continuity and flux balance by construction rather than through penalty terms in the loss. The windowing method builds the trial function from compactly supported windowed subnetworks, while the buffer method adds correction functions to enforce conditions at discrete points. Both are compared to standard soft-constrained PINNs on elliptic interface benchmarks in one and two dimensions. What the paper does well is demonstrate practical improvements. In 1D, the hard approaches consistently give better interface accuracy and avoid the tuning of loss weights that soft methods require. The windowing version reaches errors as low as 10^{-9} on structured problems, and the buffer version maintains around 10^{-5} accuracy more generally. In 2D the buffer formulation is more reliable, as windowing runs into issues with support overlaps and corners. The soft spots are mainly around the dependence on exact a priori interface geometry. Defining the windows or buffer points assumes the location is known precisely, which may not hold without approximation error for complex or curved interfaces. This could reintroduce some of the enforcement problems the methods aim to solve. The reported 2D results are less impressive than 1D, and more details on the specific network setups and training would help assess reproducibility. Readers who work with PINNs for multiphysics or interface problems in engineering and physics will find this relevant. It is worth a serious referee because the formulations are distinct and the benchmarks show clear differences from baselines, even if further validation is needed for broader use. I would recommend sending it for peer review.

Referee Report

2 major / 3 minor

Summary. The paper introduces two ansatz-based hard-constrained PINN formulations for elliptic interface problems. The windowing approach constructs the trial space from compactly supported windowed subnetworks so that interface continuity and flux balance hold by design. The buffer approach augments subnetworks with auxiliary buffer functions that enforce constraints at discrete points via a lightweight correction. Both are tested on 1D and 2D benchmarks against soft-constrained baselines, with windowing reaching O(10^{-9}) accuracy on simple 1D cases and buffer proving more robust in 2D due to reduced sensitivity to geometry details.

Significance. If the central claims hold, the work offers a practical route to enforce interface conditions exactly in PINNs without loss-weight tuning, which could improve accuracy and reliability for interface problems arising in fluid dynamics, materials science, and electromagnetics. The explicit comparison of two distinct hard-constraint mechanisms and the identification of buffer robustness in 2D provide concrete guidance for practitioners.

major comments (2)

[Abstract and §4 (Numerical Experiments)] Abstract and results presentation: reported accuracies (O(10^{-9}) in 1D windowing, O(10^{-5}) in 2D buffer) are given without network architectures, training-point counts, optimizer details, or error bars across multiple runs. This information is load-bearing for assessing whether the observed gains over soft baselines are reproducible and insensitive to hyperparameter choices.
[Windowing approach description] Windowing formulation: the claim that continuity and flux balance are satisfied exactly by construction presupposes that the interface geometry is known precisely enough to define the compactly supported windows without discretization error. For curved or non-grid-aligned interfaces, any approximation in window support or overlap introduces a coupling between constraint enforcement and the underlying discretization, undermining the asserted decoupling from PDE residual minimization. The noted 2D sensitivity to overlap and corners already signals this fragility.

minor comments (3)

[Method sections] Provide explicit formulas for the window functions and buffer corrections in the main text (rather than relegating all details to an appendix) to improve readability.
[Numerical results] Add a short discussion or table comparing the computational overhead (training time, number of parameters) of the hard-constrained formulations versus the soft baseline.
[Buffer approach] Clarify the precise definition of the discrete buffer correction points and how they are chosen when the interface is not aligned with the collocation grid.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We provide point-by-point responses to the major comments and outline the revisions we plan to implement.

read point-by-point responses

Referee: [Abstract and §4 (Numerical Experiments)] Abstract and results presentation: reported accuracies (O(10^{-9}) in 1D windowing, O(10^{-5}) in 2D buffer) are given without network architectures, training-point counts, optimizer details, or error bars across multiple runs. This information is load-bearing for assessing whether the observed gains over soft baselines are reproducible and insensitive to hyperparameter choices.

Authors: We agree with the referee that providing these implementation details is crucial for assessing reproducibility. In the revised manuscript, we will expand Section 4 to include the specific network architectures (depth and width of subnetworks), the number and distribution of training points in each subdomain, the choice of optimizer (e.g., Adam with learning rate schedule), and quantitative error statistics including means and standard deviations from at least five independent training runs with different initializations. revision: yes
Referee: [Windowing approach description] Windowing formulation: the claim that continuity and flux balance are satisfied exactly by construction presupposes that the interface geometry is known precisely enough to define the compactly supported windows without discretization error. For curved or non-grid-aligned interfaces, any approximation in window support or overlap introduces a coupling between constraint enforcement and the underlying discretization, undermining the asserted decoupling from PDE residual minimization. The noted 2D sensitivity to overlap and corners already signals this fragility.

Authors: The referee raises an important point regarding the practical realization of the windowing approach. Our theoretical construction assumes exact knowledge of the interface to define the compactly supported window functions, ensuring that the interface conditions are satisfied exactly in the continuous sense. In the discrete training, collocation points are sampled accordingly. However, for interfaces that are curved or not aligned with any underlying grid, defining the window supports exactly may indeed require numerical approximations, potentially introducing a weak coupling. This is consistent with the sensitivity to overlap parameters and corner effects that we already report in the 2D experiments, which motivated our recommendation of the buffer approach for more complex geometries. We will revise the description of the windowing method to clarify this assumption and add a brief discussion of its implications for general interface problems. revision: partial

Circularity Check

0 steps flagged

Ansatz-based hard-constrained formulations embed constraints by construction with no reduction to fitted inputs or self-referential predictions.

full rationale

The paper proposes two new formulations (windowing and buffer) that are explicitly constructed as ansatzes to satisfy interface continuity and flux balance in the trial space. These choices are presented as design decisions rather than derived predictions, and the central claims rest on empirical comparisons to soft-constrained baselines on elliptic benchmarks rather than on any self-citation chain or parameter fit that is then renamed as a result. The assumption of known interface geometry is stated up front and does not create a circular loop in the reported derivation or validation steps. This is a standard low-level ansatz construction with independent content in the numerical experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the interface is known and can be exactly represented for window or buffer construction, plus standard neural network approximation capabilities.

axioms (2)

domain assumption Interface location and geometry are known exactly a priori.
Required to construct the window functions or place buffer correction points.
standard math Neural networks can represent the sub-problems on each side of the interface with sufficient accuracy.
Implicit in the ansatz construction and residual minimization.

pith-pipeline@v0.9.0 · 5838 in / 1222 out tokens · 25857 ms · 2026-05-21T09:18:33.277781+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The windowing approach constructs the trial space from compactly supported windowed subnetworks so that interface continuity and flux balance are satisfied by design... polynomial window functions... ˜Wint(τ)=1−3τ²+2τ³
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

buffer approach... augments unrestricted subnetworks with auxiliary buffer functions that enforce boundary and interface constraints at discrete points through a lightweight correction

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 2 internal anchors

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G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nature Reviews Physics 3 (6) (2021) 422–440

work page 2021

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Raissi, P

M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational physics 378 (2019) 686–707

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S. Wang, S. Sankaran, H. Wang, P. Perdikaris, An expert’s guide to training physics-informed neural networks, arXiv preprint arXiv:2308.08468 (2023)

work page arXiv 2023

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A. D. Jagtap, G. E. Karniadakis, Extended physics-informed neural net- works (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations, Communications in Computational Physics 28 (5) (2020)

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L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, S. G. Johnson, Physics-informed neural networks with hard constraints for inverse de- sign, SIAMJournalon ScientificComputing43(6)(2021)B1105–B1132

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Sukumar, A

N. Sukumar, A. Srivastava, Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks, Com- puter Methods in Applied Mechanics and Engineering 389 (2022) 114333

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J. Wang, Y. Mo, B. Izzuddin, C.-W. Kim, Exact dirichlet boundary physics-informed neural network epinn for solid mechanics, Computer Methods in Applied Mechanics and Engineering 414 (2023) 116184

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S. Liu, H. Zhongkai, C. Ying, H. Su, J. Zhu, Z. Cheng, A unified hard- constraint framework for solving geometrically complex pdes, Advances in Neural Information Processing Systems 35 (2022) 20287–20299

work page 2022

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Sukumar, R

N. Sukumar, R. Roy, A wachspress-based transfinite formulation for exactly enforcing dirichlet boundary conditions on convex polyg- onal domains in physics-informed neural networks, arXiv preprint arXiv:2601.01756 (2026)

work page internal anchor Pith review arXiv 2026

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S. Dong, Y. Zhang, A novel method for enforcing exactly dirichlet, neu- mann and robin conditions on curved domain boundaries for physics informed machine learning, arXiv preprint arXiv:2603.21909 (2026)

work page arXiv 2026

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A. K. Sarma, S. Roy, C. Annavarapu, P. Roy, S. Jagannathan, Inter- face pinns (i-pinns): A physics-informed neural networks framework for interface problems, Computer Methods in Applied Mechanics and En- gineering 429 (2024) 117135

work page 2024

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S. Roy, C. Annavarapu, P. Roy, A. K. Sarma, Adaptive interface-pinns (adai-pinns): An efficient physics-informed neural networks framework for interface problems, arXiv preprint arXiv:2406.04626 (2024). 51

work page arXiv 2024

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Hu, T.-S

W.-F. Hu, T.-S. Lin, M.-C. Lai, A discontinuity capturing shallow neu- ral network for elliptic interface problems, Journal of Computational Physics 469 (2022) 111576

work page 2022

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S. Roy, D. R. Sarkar, C. Annavarapu, P. Roy, B. Lecampion, D. M. Valiveti, Adaptive interface-pinns (adai-pinns) for transient diffusion: Applications to forward and inverse problems in heterogeneous media, Finite Elements in Analysis and Design 244 (2025) 104305

work page 2025

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D. R. Sarkar, C. Annavarapu, P. Roy, Adaptive interface-pinns (adai- pinns) for inverse problems: Determining material properties for het- erogeneous systems, Finite Elements in Analysis and Design 249 (2025) 104373

work page 2025

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P. Roy, S. T. Castonguay, Exact enforcement of temporal continuity in sequential physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering 430 (2024) 117197

work page 2024

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Moseley, A

B. Moseley, A. Markham, T. Nissen-Meyer, Finite basis physics- informed neural networks (fbpinns): a scalable domain decomposition approach for solving differential equations, Advances in Computational Mathematics 49 (4) (2023) 62

work page 2023

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Anderson, J

R. Anderson, J. Andrej, A. Barker, J. Bramwell, J.-S. Camier, J. Cer- veny, V. Dobrev, Y. Dudouit, A. Fisher, T. Kolev, W. Pazner, M. Stow- ell, V. Tomov, I.Akkerman, J. Dahm, D.Medina, S.Zampini, MFEM: A modular finite element methods library, Computers & Mathematics with Applications 81 (2021) 42–74.doi:10.1016/j.camwa.2020.06.009

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N. Vyas, D. Morwani, R. Zhao, M. Kwun, I. Shapira, D. Brandfonbrener, L. Janson, S. Kakade, Soap: Improving and stabilizing shampoo using adam, arXiv preprint arXiv:2409.11321 (2024)

work page internal anchor Pith review Pith/arXiv arXiv 2024

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Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions, Springer, 2006

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