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Trace Regularity PINNs: Enforcing $\mathrm{H}^{\frac{1}{2}}(\partial \Omega)$ for Boundary Data

Doyoon Kim, Junbin Song

arXiv:2510.16817v2cs.LGmath.AP

TL;DR

Standard PINNs use an L2 boundary loss that may not enforce the boundary regularity needed for H^1(Ω) approximation. TRPINN uses an H^(1/2)(∂Ω) boundary loss with an essential-region discretization. The paper reports reduced H^1(Ω) error and theoretical H^1(Ω) convergence when the loss tends to zero.

  • Problem

    An L2 boundary loss may leave boundary conditions insufficiently enforced, allowing discrepant boundary data to alter the effective problem and affect the interior approximation.

    Source: §1 · pp. 1–2

  • Method

    TRPINN incorporates the H^(1/2)(∂Ω) boundary norm using an essential-region discretization that avoids denominator evaluations.

    Source: §1 · p. 2

  • Results

    TRPINN reduces H^1(Ω) error by one to two orders of magnitude over vanilla PINN on problems with challenging boundary conditions.

    Source: §8 · p. 14

  • Takeaways & Limitations

    When its loss converges to zero, TRPINN’s approximation converges to the exact solution in H^1(Ω), unlike the stated guarantee for vanilla PINN.

    Source: §8 · p. 15

    The stated model assumes a bounded Lipschitz domain and boundary data g in H^(1/2)(∂Ω).

    Source: §2.2 · p. 3

Abstract

from arXiv · show

We propose an enhanced physics-informed neural network (PINN), the Trace Regularity Physics-Informed Neural Network (TRPINN), which enforces the boundary loss in the Sobolev-Slobodeckij norm $H^{1/2}(\partial \Omega)$, the correct trace space associated with $H^1(\Omega)$. We reduce computational cost by computing only the theoretically essential portion of the semi-norm and enhance convergence stability by avoiding denominator evaluations in the discretization. By incorporating the exact $H^{1/2}(\partial \Omega)$ norm, we show that the approximation converges to the true solution in the $H^{1}(\Omega)$ sense, and, through Neural Tangent Kernel (NTK) analysis, we demonstrate that TRPINN can converge faster than standard PINNs. Numerical experiments on the Laplace equation with highly oscillatory Dirichlet boundary conditions exhibit cases where TRPINN succeeds even when standard PINNs fail, and show performance improvements of one to three decimal digits.

1. Introduction

Standard PINNs commonly use an L2 boundary loss, although H^(1/2)(∂Ω) is the trace space associated with H^1(Ω). TRPINN incorporates that boundary regularity with an essential-region, denominator-free discretization and provides theoretical and NTK-based convergence results.

1. Introduction

  • The paper frames PINNs as neural-network approximations that embed PDE residuals and boundary or initial conditions in a loss function.Source: §1 · p. 1
  • Standard L2 boundary losses may not control the regularity needed for H^1(Ω) approximation, while imposing H^1(∂Ω) can require unavailable boundary derivatives.For u in H^1(Ω), the trace data are guaranteed only in H^(1/2)(∂Ω).Source: §1 · pp. 1–2
  • TRPINN computes only essential regions of the H^(1/2)(∂Ω) seminorm using an adapted trapezoidal rule that removes denominator evaluations and does not require Dg.The method is described as having comparable training time for the same iteration count, with faster convergence in some cases due to early stopping.Source: §1 · p. 2
  • TRPINN’s H^(1/2)(∂Ω) boundary loss makes convergence of the loss to zero imply convergence to the exact solution in H^1(Ω), and its NTK analysis shows larger eigenvalues.The paper relates larger eigenvalues to improved performance.Source: §1 · p. 2

2. Theoretical Framework

The framework uses boundary trace spaces and Sobolev norms to motivate the TRPINN loss. Its model problem is a divergence-form elliptic equation with stated domain, coefficient, and boundary-data assumptions.

2. Theoretical Framework

  • The framework identifies boundary trace spaces as the basis for the loss design and the estimates used in later analysis.Source: §2 · p. 3

2.1. Function Spaces and Norms.

  • H^k(Ω) comprises functions with weak derivatives through order k in L2(Ω), while H^(1/2)(∂Ω) and H^(3/2)(∂Ω) are Sobolev-Slobodeckij spaces on the boundary.Source: §2.1 · p. 3
  • The indicator in the seminorm restricts integration to its support, absorbs the boundary L2 norm into the existing loss, and adds the remaining seminorm as a training-loss component.Source: §2.1 · p. 3

2.2. Model Problem and Assumptions.

  • The model is a divergence-form elliptic PDE on a bounded Lipschitz domain, with boundary data g in H^(1/2)(∂Ω) and coefficients subject to the stated regularity assumptions.The exposition assumes zero lower-order coefficients and zero forcing; the paper says nonzero forcing is deferred to future work.Source: §2.2 · pp. 3–4
  • For the stated problem, the solution lies in H^1(Ω), whose trace lies in H^(1/2)(∂Ω), making that boundary norm necessary and sufficient for the formulation.Source: §2.2 · p. 4
  • The formulation also extends to non-divergence operators with H^2(Ω) solutions and H^(3/2)(∂Ω) boundary data, with the seminorm discretization applied to Dg.Source: §2.2 · p. 4

3. Discretization of Semi-Norm

The discretization targets the H^(1/2)(∂Ω) boundary seminorm rather than relying only on an L2 boundary loss. It restricts integration to essential regions and uses a scheme that avoids denominator evaluation while reducing computational cost.

3. Discretization of Semi-Norm

  • The boundary loss targets H^(1/2)(∂Ω) because standard PINNs typically use L2(∂Ω), while direct Monte Carlo evaluation of the fractional norm may be costly and error-prone.The fractional norm is presented as a more rigorous and physically consistent way to enforce boundary conditions.Source: §3 · p. 4
  • The method extends the single trapezoidal rule to the double integral and applies it to discretize the H^(1/2) seminorm using sampled boundary points.The boundary samples are ordered, and δ is chosen sufficiently small relative to the sampling points.Source: §3 · p. 4
  • TRPINN’s restricted integration region reduces seminorm computation from the full-region O(N^2) cost to O(N), and cancellation avoids evaluating the denominator.Boundary computation adds little to overall training cost because interior residuals dominate; Table 4 is reported to show no material difference.Source: §3 · p. 5
  • The TRPINN integration region fully covers the support of |x−y|<δ, whereas the DRM-AQR region leaves part uncovered; both avoid computing over all of ∂Ω×∂Ω.Source: §3 · p. 5, Figure 1 · p. 5

4. Trace Regularity Physics-informed neural networks

Conventional PINNs combine interior residual and boundary losses, but their L2 boundary loss and sensitivity to loss weights can limit performance. TRPINN adds a discretized H1/2 boundary semi-norm to enforce the theoretically exact boundary norm, with computational cost comparable to standard PINNs.

4.1. Conventional PINN.

  • Conventional PINNs train a neural network using interior PDE residuals and boundary-condition terms in a loss function.The interior collocation points lie in Ω, while the boundary points lie on ∂Ω; α and β weight their respective losses.Source: §4.1 · p. 5
  • Conventional PINNs can be sensitive to loss weights, and their L2 boundary loss may not capture the exact regularity of boundary data.Weight sensitivity can bias training toward boundary data, underfit the interior residual, and slow or destabilize convergence, particularly for high-frequency solutions.Source: §4.1 · p. 6

4.2. Trace Regularity PINN (TRPINN).

  • TRPINN augments the interior and boundary losses with a discretized H1/2 semi-norm, enforcing the theoretically exact boundary norm.Its loss weights the three terms by α, β, and γ.Source: §4.2 · p. 6
  • The semi-norm computation restricts work to theoretically necessary regions and avoids denominator evaluations, keeping cost comparable to a standard PINN and improving loss-calculation stability.The discretization uses the semi-norm described in (3.5).Source: §4.2 · p. 6

5. H1-convergence in the interior Ω

This section argues that enforcing the H1/2 boundary norm yields H1 convergence in the interior when the loss tends to zero. It contrasts this guarantee with conventional L2 boundary enforcement and discusses boundary data outside H1, including an oscillatory case where vanilla PINN fails.

5. H1-convergence in the interior Ω

  • If TRPINN's loss converges to zero, its approximation is guaranteed to converge to the ground-truth solution in H1(Ω).The guarantee follows because TRPINN enforces the right-hand side of (5.3).Source: §5 · p. 6
  • Conventional PINNs enforce only L2(∂Ω), which does not guarantee H1 convergence in the domain even when the loss tends to zero.The passage notes prior examples where conventional PINNs fail to achieve interior H1 convergence.Source: §5 · p. 6
  • A PINN penalizing the H1 boundary norm cannot properly handle data that belong to H1/2(∂Ω) but not H1(∂Ω).The stated boundary regularity follows from the trace theorem for the considered divergence-form equations.Source: §5 · p. 6
  • For oscillatory boundary data with frequency V = 7, the vanilla PINN completely fails to converge, while the discussion identifies H1/2 as the enforceable boundary regularity.The passage frames V as a positive integer representing oscillation frequency.Source: §5 · p. 7
  • TRPINN does not require auxiliary boundary-derivative data Dg, and its semi-norm discretization keeps overall computational cost comparable to a standard PINN.The passage notes that using Dg would require additional neural-network auto-differentiations.Source: §5 · p. 7

6. Neural Tangent Kernel

The NTK analysis compares training dynamics through eigenvalues, which indicate convergence rates at individual points. It argues that TRPINN has larger eigenvalues and faster convergence than a conventional PINN, and reports a computed spectrum comparison across three boundary-sampling methods.

6. Neural Tangent Kernel

  • The paper formalizes TRPINN's gradient flow from the NTK perspective to analyze its training dynamics.The NTK is presented as a tool for analyzing neural-network training dynamics.Source: §6 · p. 7

6.1. Comparative analysis of PINN and TRPINN from the NTK viewpoint.

  • The comparative analysis frames the H1/2 boundary discrepancy as a replacement for the standard L2 boundary discrepancy.It presents the resulting larger eigenvalues and faster convergence as relevant to oscillatory and ill-conditioned regimes.Source: §6.1 · p. 8
  • The NTK analysis states that TRPINN converges faster than standard PINNs, with larger eigenvalues indicating higher pointwise convergence rates.The analysis says the increase occurs primarily among small eigenvalues, alleviating disparities in speed across points.Source: §6.1 · pp. 8–9

6.2. Computed NTK Spectrum on boundary ∂Ω.

  • In the computed boundary spectrum, TRPINN's eigenvalues are larger than PINN's across linspace, randomized, and uniform-random sampling.The experiment compares the top 100 eigenvalues using a three-hidden-layer network of width 128 and identical loss scaling coefficients β = γ = 100.Source: §6.2 · p. 9, Figure 2 · p. 9

7. Numerical Experiments

The experiments compare vanilla PINN and TRPINN on Laplace problems with highly oscillatory and sharp-peaked boundary data, using harmonic extensions as exact solutions. Across these settings, TRPINN generally performs better and is less sensitive to weight choices.

7. Numerical Experiments

  • The experiments use the unit disk, harmonic-extension ground truth, and best relative H1 error observed during Adam plus L-BFGS training to compare the models.For the high-oscillation case, V=20; the sharp-peaked case uses V=7 with other hyperparameters unchanged.Source: §7 · p. 10, §7.1 · p. 10, §7.2 · p. 13

7.1. Boundary Condition with High Oscillations.

  • For sin(20θ), vanilla PINN relative errors are approximately 1, while TRPINN reaches errors at the first and third decimal places and improves over vanilla by up to three decimal places.Even a 1% semi-norm share captures part of the solution, and increasing that share brings predictions closer to the true boundary values.Source: §7.1 · p. 10
  • Across nearly all tested model sizes, relative errors differ by approximately one to two decimal places, with comparable Adam costs and vanilla PINN taking longer during L-BFGS despite underperforming TRPINN.TRPINN learns substantially during Adam and improves further with L-BFGS; vanilla PINN shows minimal Adam learning before limited L-BFGS improvement and early stopping.Source: §7.1 · p. 12

7.2. Boundary Condition with Multiple Sharp Peaks.

  • For sharp-peaked boundary data, TRPINN improves relative L2 error by two decimal places and relative H1 error by one, while vanilla PINN errors remain near 1.Results are broadly consistent across weight choices; boundary predictions from vanilla PINN are nearly zero, while TRPINN predictions are nearly accurate.Source: §7.2 · pp. 13–14

8. discussion

The discussion attributes TRPINN’s gains to enforcing the theoretically consistent boundary regularity and capturing oscillations and sharpness. It presents H1/2 boundary control as necessary for guaranteeing convergence in H1.

8. discussion

  • On challenging boundary conditions, TRPINN improves H1 error by one to two orders of magnitude over vanilla PINN.The discussion attributes the gain to enforcing theoretically consistent boundary regularity and capturing boundary oscillation and sharpness.Source: §8 · p. 14
  • Computing only the essential H1/2 semi-norm portion keeps TRPINN’s computational cost comparable to vanilla PINN, while its weights can be chosen more robustly.TRPINN also uses boundary data g without requiring its derivative Dg.Source: §8 · pp. 14–15
  • With the H1/2 boundary norm, loss convergence to zero implies convergence to the exact solution in H1; vanilla PINN has no such guarantee.The discussion characterizes H1/2 boundary control as theoretically appropriate rather than excessive.Source: §8 · p. 15

9. conclusion

TRPINN enforces the essential H^1/2(∂Ω) boundary norm, reducing H^1(Ω) error and offering theoretical convergence guarantees. The conclusion also reports faster convergence and greater robustness to boundary-loss weights.

9. conclusion

  • TRPINN selectively enforces essential H^1/2(∂Ω) boundary components with almost no added boundary-loss computational cost.The paper reports reduced H^1(Ω) relative error, H^1(Ω) convergence when the loss tends to zero, faster convergence through mitigation of the NTK per-mode learning-speed gap, and greater robustness to boundary-loss weights.Source: §9 · p. 15
  • The authors identify extension to sources f ∈ H^-1(Ω) and time-dependent equations as ways to broaden TRPINN's applicability.Source: §9 · p. 15

Appendix A. Approximation solutions using approximate coefficients

The appendix addresses divergence-form PDEs whose operators may not map into L2(Ω), motivating approximation through problems with approximate coefficients. It states that sufficiently small ε yields a solution close to the original in H^1(Ω).

Appendix A. Approximation solutions using approximate coefficients

  • For general coefficients a_ij ∈ L∞(Ω) with f_i = 0, the appendix proposes approximating the PDE with a problem using approximate coefficients.This addresses the fact that the divergence-form operator need not map into L2(Ω), especially without coefficient regularity.Source: Appendix A · pp. 15–16
  • The approximate coefficients retain the same ellipticity as a_ij.Source: Appendix A · p. 16
  • Problem (A.1) has a unique solution uε ∈ H1(Ω); ε is chosen sufficiently small so this solution is close to the true solution u of (2.3) in the H1(Ω) sense.Source: Appendix A · p. 16
  • The H^1(Ω) closeness result compares solutions with the same forcing function and boundary data, with a constant C independent of ε.The lemma is stated to remain valid when f_i ≠ 0.Source: Appendix A · p. 16
  • For coefficients a_ij ∈ L∞(Ω), the appendix states that sufficiently small ε makes the approximate problem usable to obtain an approximate solution.The stated justification is that the left-hand side of (A.2) tends to zero as ε → 0.Source: Appendix A · p. 16
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