Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs

TL;DR

Gated X-TFC is proposed, a novel framework for both forward and inverse problems, that overcomes limitations through a soft, learned domain decomposition and introduces an operator-conditioned meta-learning layer that learns a probabilistic mapping from PDE parameters to optimal gate configurations, enabling fast, uncertainty-aware warm-starting for new problem instances.

Abstract

Physics-informed neural networks (PINNs) and related methods struggle to resolve sharp gradients in singularly perturbed boundary value problems without resorting to some form of domain decomposition, which often introduce complex interface penalties. While the Extreme Theory of Functional Connections (X-TFC) avoids multi-objective optimization by employing exact boundary condition enforcement, it remains computationally inefficient for boundary layers and incompatible with decomposition. We propose Gated X-TFC, a novel framework for both forward and inverse problems, that overcomes these limitations through a soft, learned domain decomposition. Our method replaces hard interfaces with a differentiable logistic gate that dynamically adapts radial basis function (RBF) kernel widths across the domain, eliminating the need for interface penalties. This approach yields not only superior accuracy but also dramatic improvements in computational efficiency: on a benchmark one dimensional (1D) convection-diffusion, Gated X-TFC achieves an order-of-magnitude lower error than standard X-TFC while using 80 percent fewer collocation points and reducing training time by 66 percent. In addition, we introduce an operator-conditioned meta-learning layer that learns a probabilistic mapping from PDE parameters to optimal gate configurations, enabling fast, uncertainty-aware warm-starting for new problem instances. We further demonstrate scalability to multiple subdomains and higher dimensions by solving a twin boundary-layer equation and a 2D Poisson problem with a sharp Gaussian source. Overall, Gated X-TFC delivers a simple alternative alternative to PINNs that is both accurate and computationally efficient for challenging boundar-layer regimes. Future work will focus on nonlinear problems.

Figures (7)

• Figure 1: Distribution of training, validation and test collocation points.
• Figure 2: Gated X--TFC across decreasing diffusion. Left column: optimization target (pointwise PDE residual) and absolute error shown only for assessment; no interface penalties are used. Right column: learned RBF width field and logistic gate; the dashed vertical line marks the learned split $x_s^\ast$.
• Figure 3: Gated X--TFC solution (solid) vs. exact (dashed) for $\nu=[10^{-2},10^{-3},10^{-4}]$. The dashed vertical line marks the learned split $x_s^\ast$; the inset zooms the boundary layer near $x=1$. The exact solution is used only for assessment—training minimizes the strong-form residual with no interface penalties.
• Figure 4: Gated X--TFC inverse solutions across decreasing diffusion. Red: posterior mean; shaded: $95\%$ model band; blue: data; black dashed: exact (for reference only).
• Figure 5: Bayesian optimization (BO) trace for the inverse problem ($\nu_{\text{true}}=5\times 10^{-3}$, $\nu_{\text{pred}}=5.37\times 10^{-3}$). Top-left: log-evidence at each evaluation (dotted) and its best-so-far envelope (solid); the star marks the final maximum-evidence model. Top-right/bottom: best-so-far hyperparameters aligned with the envelope—$\log_{10}\nu$, split $x_s$, and gate scale $\varepsilon_{\mathrm{scale}}$. BO maximizes evidence; the starred parameters are used for the reported reconstruction.