Consistent CutPINNs for Elliptic PDEs on Curved Level-Set Domains

Technical Summary: Consistent CutPINNs for Elliptic PDEs on Curved Level-Set Domains

Problem Statement
Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving partial differential equations (PDEs), making them naturally suited for domains defined implicitly by level-set functions. However, standard PINN formulations suffer from a theoretical inconsistency when applied to second-order elliptic problems. The standard loss function penalizes the boundary mismatch in the $L^2(\partial\Omega)$ norm. While the well-posedness of elliptic problems is governed by the $H^1(\Omega)$ energy norm, the relevant a priori estimates require control of the boundary residual in the fractional Sobolev space $H^{1/2}(\partial\Omega)$. Since the embedding $H^{1/2}(\partial\Omega) \subset L^2(\partial\Omega)$ is strict, minimizing the standard $L^2$ boundary loss does not guarantee the minimization of the $H^1$ error. This discrepancy allows for high-frequency oscillations that vanish at collocation points but possess large $H^{1/2}$ seminorms, leading to poor convergence rates and instability, particularly in "cut-cell" configurations where the domain boundary intersects a sampling grid.

Previous work by Bonito et al. [4] addressed this by introducing "consistent" loss functions that approximate the $H^{1/2}(\partial\Omega)$ norm via a discrete surrogate. However, their construction relies on the Kuhn–Tucker simplicial decomposition of flat boundary faces (specifically on the unit cube) and the affine structure of those faces. This approach does not extend to smooth, curved boundaries defined by level-set functions, leaving a gap in the theoretical and practical application of consistent PINNs to general curved domains.

Methodology
The paper proposes the Consistent CutPINN framework, designed specifically for bounded two-dimensional domains $\Omega = \{x \in \mathbb{R}^2 : \phi(x) < 0\}$ where $\phi$ is a $C^2$ level-set function. The core methodology involves constructing a discrete surrogate for the $H^{1/2}(\partial\Omega)$ norm directly from collocation points on the curved boundary, without requiring an underlying finite element mesh or simplicial decomposition.

1. Discrete Norm Construction: The authors define a discrete $H^{1/2}$ norm using a double-sum discretization of the intrinsic Gagliardo seminorm over collocation points $Z = \{z_1, \dots, z_m\} \subset \partial\Omega$. The kernel uses the ambient Euclidean distance $|z - z'|$ rather than arc-length distance.
2. Theoretical Foundation: The framework relies on a Chord-arc lemma (Lemma 1), which establishes that for a $C^2$ curve, the Euclidean chord distance is comparable to the arc-length distance. This equivalence allows the authors to prove a two-sided norm equivalence (Theorem 3) between the discrete surrogate and the continuous $H^{1/2}(\partial\Omega)$ trace norm, provided the boundary points are equidistant in arc length (or quasi-uniform).
3. Loss Functions: Two consistent loss functions are defined:
   • $L^*_{sq,2}$: Uses the standard $L^2$ interior residual and the new discrete $H^{1/2}$ boundary residual.
   • $L^*_{sq,\gamma}$: Uses an $L^\gamma$ interior residual (with $\gamma \to 1$) to better approximate the $H^{-1}$ norm, combined with the discrete $H^{1/2}$ boundary term.
4. Optimal Recovery: The framework is analyzed within the optimal recovery theory under Besov regularity assumptions. The authors derive an a priori $H^1$ error bound (Theorem 7) showing that minimizing the consistent loss controls the $H^1$ error at the optimal recovery rate.

Key Contributions

• Extension to Curved Domains: The paper bridges the gap between consistent loss functions and implicitly defined curved boundaries. It demonstrates that the discrete $H^{1/2}$ norm constructed from Euclidean distances on a $C^2$ curve is norm-equivalent to the continuous trace norm, removing the reliance on flat-face simplicial decompositions.
• A Priori Error Bounds: The authors establish a rigorous a priori $H^1$ error bound for the Consistent CutPINN framework. The bound depends on the maximum curvature of the boundary through the Chord-arc constant and achieves optimal convergence rates under Besov regularity.
• Robustness to Cut-Cell Configurations: Theoretical analysis and numerical experiments show that the consistent loss is significantly more robust to the position of the domain relative to the sampling points compared to standard PINNs.
• Probabilistic Robustness: The paper includes a probabilistic analysis (Proposition 4) showing that the discrete seminorm concentrates around its mean even under i.i.d. Monte Carlo boundary sampling, suggesting the method's effectiveness extends beyond strictly equidistant sampling.

Results
Numerical experiments were conducted on a disk (convex) and a non-convex flower domain (five-fold symmetry) using manufactured solutions.

• Accuracy: The consistent loss $L^*_{sq,2}$ consistently outperformed the standard PINN loss ($L_{sq,1}$). On the disk, it achieved 4.9 times lower $H^1$ error at high resolution; on the flower domain, the improvement factor reached 8.1 times.
• Convergence Rates: The standard loss saturated at a sub-optimal error level (slope $\approx -0.31$) because the $L^2$ boundary term failed to control the $H^{1/2}$ trace. In contrast, the consistent loss $L^*_{sq,2}$ achieved the predicted optimal convergence slope of $\approx -0.49$ (matching the theoretical $-0.50$).
• Stability: In a "moving disk" experiment where the domain center was shifted 121 times, the standard loss exhibited high variance and occasional divergence (spikes in error). The consistent loss $L^*_{sq,2}$ reduced the standard deviation of the error by a factor of 26 and remained stable in 120 out of 121 configurations.
• Non-Uniform Sampling: Even when boundary points were sampled via i.i.d. Monte Carlo (violating the equidistant assumption of the main theorem), the consistent loss still reduced the $H^1$ error by a factor of 3.9 compared to the standard loss.

Significance
The paper claims that Consistent CutPINN provides the first framework that combines consistent loss functions with implicitly defined curved domains. By proving that the discrete $H^{1/2}$ surrogate built from Euclidean distances is norm-equivalent to the continuous trace on $C^2$ curves, the work validates the use of consistent losses in mesh-free settings for complex geometries. The results demonstrate that addressing the norm mismatch in the loss function is critical for achieving optimal convergence rates and robustness in PINNs, particularly when the solution exhibits boundary layers or high-frequency oscillations. The authors note that the current analysis is restricted to $d=2$ due to the reliance on global arc-length parametrization for the Chord-arc lemma, leaving extensions to 3D and domains with corners as future work.