Trace Regularity PINN (TRPINN)

• TRPINN is an advanced physics-informed neural network that enforces boundary regularity using the Sobolev–Slobodeckij H¹/² semi-norm.
• It improves error bounds, stability, and convergence in solving elliptic PDEs with complex or oscillatory boundary data.
• Efficient strategies like boundary localization and trapezoidal rule discretization reduce computational complexity from O(N²) to O(N).

The Trace Regularity Physics-Informed Neural Network (TRPINN) is an advanced variant of physics-informed neural networks that enforces rigorous boundary regularity in the solution of partial differential equations (PDEs). TRPINN incorporates the Sobolev–Slobodeckij $H^{1/2}(\partial\Omega)$ semi-norm to constrain neural network boundary outputs, aligning numerical treatment with the true trace space of underlying energy norms. This approach yields improved error bounds, stability, and convergence particularly in the presence of complex or highly oscillatory boundary data.

1. Theoretical Motivation: Boundary Regularity and Trace Spaces

Standard PINNs enforce boundary conditions through minimization of the $L^2$-norm of the discrepancy between network predictions and prescribed boundary data. However, for elliptic PDEs, the mathematically appropriate trace space for solutions in the energy space $H^1(\Omega)$ is $H^{1/2}(\partial\Omega)$. Conventional $L^2$ losses do not guarantee control over the solution’s smoothness or oscillatory components on the boundary, leading to suboptimal convergence in the interior ($H^1$ norm) and possible failure in capturing features such as high-frequency boundary trends.

The trace theorem establishes that for $u \in H^1(\Omega)$, the restriction $u|_{\partial\Omega} \in H^{1/2}(\partial\Omega)$. Therefore, enforcing a boundary loss in $H^{1/2}(\partial\Omega)$ is essential for consistent numerical approximation theory and for guaranteeing energy-norm convergence of learned solutions. TRPINN is designed to meet this requirement by embedding this regularity directly in its loss function.

2. Mathematical Formulation: Sobolev–Slobodeckij Norm in Boundary Loss

The core innovation in TRPINN is the use of the Sobolev–Slobodeckij norm for boundary loss. The boundary loss between neural network output $g_{nn}$ and boundary data $g$ is given as: $$\|g_{nn} - g\|_{H^{1/2}(\partial\Omega)}^2 = \|g_{nn} - g\|_{L^2(\partial\Omega)}^2 + [g_{nn} - g]_{H^{1/2}(\partial\Omega)}^2$$ where the semi-norm is defined by: $$[g]_{H^{1/2}(\partial\Omega)}^2 = \iint_{\partial\Omega \times \partial\Omega} \frac{|g(x) - g(y)|^2}{|x - y|^2} \, dS(x) \, dS(y)$$ This formulation ensures that both pointwise error and smoothness/relative differences across the boundary are minimized. Theoretical analysis confirms that enforcing the $H^{1/2}$ norm leads to convergence in the $H^1(\Omega)$ sense for the solution, promoting more accurate and physically faithful solutions in elliptic and related boundary value problems.

3. Computational Strategy and Efficiency

Direct calculation of the $H^{1/2}$ semi-norm involves a computationally expensive double integral ($O(N^2)$ operations) and susceptibility to numerical instability, especially in evaluating denominators for small pairwise distances. TRPINN introduces two primary efficiency strategies:

1. Boundary Localization: The integral is restricted to a “theoretically essential subset”—only point pairs $(x, y)$ with $|x - y| \leq \delta$ for small $\delta$ are considered. This localization targets the dominant contribution to the semi-norm and significantly reduces computational load.
2. Efficient Discretization: A single trapezoidal rule is used, focusing on neighbor differences, e.g., $\sum_{i=1}^{N} |g(x_{i+1}) - g(x_i)|^2$ and similar terms, reducing complexity to $O(N)$ and minimizing instability from small denominators.

Avoiding explicit denominator evaluation for every pair further stabilizes the numerical integration. These measures ensure that TRPINN maintains computational parity with standard PINNs while achieving stricter boundary regularity enforcement.

4. Training Dynamics and Convergence: Neural Tangent Kernel Perspective

The training behavior of PINNs can be theoretically characterized by Neural Tangent Kernel (NTK) analysis, which connects the network’s convergence rate to the spectrum of the corresponding kernel matrix derived from loss gradients. For standard PINNs with $L^2$ boundary loss, NTK matrices display numerous small eigenvalues, particularly in high-frequency solution components. This results in slow convergence and can impede the learning of complex boundary behavior.

TRPINN modifies the NTK by introducing a symmetric positive definite matrix $M$ in the analysis of the boundary penalty. This adjustment yields larger eigenvalues in the boundary-influenced modes, enhancing the convergence rate during gradient descent and making slow-learning modes more accessible to the optimizer. Both theoretical derivations and numerical evidence demonstrate that TRPINN achieves more rapid and stable convergence compared to vanilla PINNs, even for highly oscillatory solutions.

5. Numerical Performance: Experiments on Oscillatory Boundary Data

Experiments described in (Kim et al., 19 Oct 2025) validate TRPINN’s theoretical foundation using the Laplace equation on domains with highly oscillatory Dirichlet boundary conditions (e.g., $g(\theta) = \sin(V\theta)$ with large $V$ on the unit disk). Standard PINNs often fail to learn such solutions, converging to trivial outputs and exhibiting high relative error ($\sim 1$ in both $H^1$ and $L^2$ norms). TRPINN, even with moderate enforcement of the $H^{1/2}$ semi-norm, accurately captures the oscillatory boundary and yields correct harmonic extensions throughout the interior. Relative errors fall by one to three decimal digits relative to standard PINNs, confirming robustness and improved accuracy for complex boundary scenarios.

6. Implications for Error Bounds, Stability, and Applicability

The overall error in PINN approximations comprises the sum of approximation, generalization, and optimization errors, all of which depend crucially on solution regularity and trace properties (Ryck et al., 2024). When trace error is minimized, as in TRPINN, stability results such as

$$\|u - u_\theta\|_{L^2(\Omega)} \leq C\left(\|L[u_\theta] - f\|_{L^2(\Omega)} + \|u_\theta - g\|_{L^2(\partial\Omega)}\right)$$

are sharpened. A plausible implication is that with rigorous enforcement of trace regularity, stability constants decrease, approximation bounds sharpen, and trained models exhibit lower optimization error due to better conditioning of the associated system matrices. This also enhances performance on inverse problems and reduces “spectral bias” at the boundary, further improving training dynamics.

7. Context in PINN Research and Future Directions

TRPINN represents a mathematically grounded enhancement in the PINN paradigm, elevating boundary treatment to match the regularity properties required by elliptic and related PDEs. This alignment yields superior performance in problems with challenging boundary data, and does not require increased data sampling or boundary derivative observations. The integration of efficient numerical methods (single-application trapezoidal rule and localized integration) broadens TRPINN’s practicality for large-scale and high-dimensional problems. These attributes suggest TRPINN is more robust with respect to loss weighting and model selection, and a plausible direction is further exploration of network architectures and sampling methods that complement trace-regularity enforcement. This suggests TRPINN is positioned to address limitations in standard PINNs across scientific, mathematical, and engineering applications requiring strong guarantees of solution regularity.