Technology Category
Machine Learning: Transfer, Domain Adaptation, Multi-Task LearningConstraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Mixed Discrete/Continuous Search
Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs
🤖 AI Summary
To address the challenges in singularly perturbed problems—namely, the inability of Physics-Informed Neural Networks (PINNs) to resolve sharp boundary layers and the reliance on manually designed interface penalties in domain decomposition—this paper proposes a Soft Domain Decomposition PINN (SDD-PINN). Our method introduces three key innovations: (1) a learnable logical gate enabling differentiable, dynamic soft domain partitioning, eliminating hard interfaces and explicit penalty terms; (2) an operator-conditioned meta-learning layer for parameter-driven, uncertainty-aware initialization; and (3) integration of the X-TFC framework with probabilistic mapping to adaptively tune RBF kernel widths, enhancing boundary-layer resolution. On 1D convection–diffusion problems, SDD-PINN achieves one-order-of-magnitude error reduction, 80% fewer collocation points, and 66% faster training. The approach generalizes successfully to multi-region and 2D Poisson problems, demonstrating broad applicability and robustness.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Resolving sharp gradients in singularly perturbed PDEs
Eliminating interface penalties in domain decomposition
Improving computational efficiency for boundary layer problems
Innovation
Methods, ideas, or system contributions that make the work stand out.
Soft learned domain decomposition with differentiable logistic gate
Dynamic adaptation of RBF kernel widths across domain
Operator-conditioned meta-learning for probabilistic gate configuration mapping
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