enforce boundary conditions

Physical-informed neural networks (PINNs) suffer from inaccurate enforcement of Dirichlet boundary conditions, leading to large boundary errors and unstable convergence. To address this, we propose a hybrid FEM-PINN framework that embeds the finite element method (FEM) within a boundary-adjacent region to construct a hard boundary constraint mechanism—enabling the first-ever strong imposition of Dirichlet conditions in PINNs. This approach departs from conventional soft-constraint paradigms while preserving strict physical consistency with the underlying PDEs. Evaluated across six PDE benchmarks with progressively increasing geometric complexity, our method reduces boundary errors by one to two orders of magnitude, accelerates convergence, and enhances training robustness. The framework demonstrates strong generalizability to industrial-scale PDE problems involving complex geometries.

本文提出一种新的架构，在保持现有核积分神经算子表达能力的同时，直接满足齐次Dirichlet边界条件，适用于任意网格数据和一般几何形状。

This work addresses the challenge that existing physics-informed machine learning methods struggle to enforce Dirichlet, Neumann, and Robin boundary conditions exactly on arbitrary curved quadrilateral domains, particularly due to compatibility constraints at corners where Neumann and Robin boundaries intersect. To overcome this limitation, the authors propose a systematic framework that integrates exact geometric mapping, the Theory of Functional Connections (TFC), and transfinite interpolation to construct trial functions that rigorously satisfy all boundary conditions and vertex compatibility requirements. These trial functions are embedded within an Extreme Learning Machine (ELM) to solve partial differential equations. The method achieves machine-precision enforcement of boundary conditions on complex curved quadrilateral domains—surpassing conventional approaches that only approximate such constraints—and demonstrates high accuracy and broad applicability across a range of linear/nonlinear and steady/unsteady problems.

Addressing the challenge of robustly enforcing mixed Dirichlet/Neumann/Robin boundary conditions on complex 3D geometries in Physics-Informed Neural Networks (PINNs), this work systematically benchmarks existing boundary enforcement techniques and introduces the first geometry-agnostic, PDE-agnostic, and boundary-type-agnostic unified PINNs framework. The proposed framework integrates geometry-aware sampling, dynamically weighted boundary loss, and adaptive constraint embedding—operating without mesh dependence and fully compatible with strong-form PDE formulations. Comprehensive evaluation across multiple nontrivial 3D benchmark problems demonstrates a 30–50% reduction in boundary error compared to state-of-the-art methods, significantly reduced hyperparameter sensitivity, and markedly improved robustness and generalization. This work provides critical methodological foundations for advancing PINNs toward engineering-grade reliability as numerical solvers.

This work addresses the safety-critical boundary control problem for unknown partial differential equation (PDE) dynamical systems. We propose Neural Boundary Control Barrier Functions (BCBFs), the first framework to integrate safety filtering into PDE boundary control. BCBFs explicitly model the mapping from boundary inputs to system outputs and exhibit linear dependence on control inputs, enabling seamless embedding into real-time quadratic programming (QP)-based safety filters for model-agnostic online safety enforcement. The approach unifies treatment of hyperbolic, parabolic, and Navier–Stokes-type PDEs without requiring exact system models, significantly improving adherence of boundary outputs to user-specified safety constraints. Experimental results demonstrate broad applicability, plug-and-play deployment, and superior performance over existing baselines.

This work addresses key limitations of conventional physics-informed neural networks (PINNs)—including slow convergence, sensitivity to loss weighting, and difficulty in accurately enforcing boundary conditions—by introducing a unified hard-soft PINN (HSPINN) framework. The method explicitly enforces Dirichlet and periodic boundary conditions as hard constraints through analytical or polynomial lifting, masking functions, and periodic feature mappings. Meanwhile, the PDE residuals, Neumann fluxes, and initial conditions are incorporated as soft constraints. To eliminate manual hyperparameter tuning, an inverse shared Softmax strategy is employed to adaptively balance the weights of all loss components. The proposed approach demonstrates significantly improved convergence speed, accuracy, and numerical stability across elliptic (Poisson), parabolic (Burgers), and hyperbolic (periodic convection) problems without requiring user-specified loss weights.

This work addresses the challenge of achieving high-order Sobolev accuracy in physics-informed neural networks (PINNs) for elliptic Dirichlet boundary value problems. The authors propose boundary-adaptive PINNs that exactly embed Dirichlet conditions into the network output by multiplying with a first-order normalized smooth boundary distance function ρ. Through a combined analysis grounded in approximation theory and statistical learning, they establish—for the first time—that merely satisfying boundary conditions is insufficient to guarantee an $H^2(\Omega)$ error bound, and rigorously identify the necessity of the first-order normalization property for ρ. Leveraging ReQU or tanh activation functions, they derive novel VC-dimension bounds for derivative hypothesis spaces and high-order Sobolev approximation rates for shallow networks, leading to a provable $H^2$ priori error estimate. Numerical experiments confirm that a properly constructed ρ significantly enhances both accuracy and convergence in MET computations.

This work addresses the challenge of efficiently and accurately solving parametrized partial differential equations (PDEs) under varying boundary conditions, a task where traditional reduced-order models often struggle. To overcome this limitation, the authors propose the Graph-Instructed Neural Network (GINN), a novel approach that, for the first time, integrates graph structures into a deep learning framework to directly learn the mapping between parametric descriptions of the computational domain and the corresponding PDE solutions. By abandoning conventional Galerkin projections and fully connected architectures, GINN leverages graph representations to flexibly accommodate variable boundary conditions without requiring re-discretization for each new configuration. Experimental results demonstrate that GINN significantly outperforms traditional methods across multiple scenarios, achieving high accuracy while exhibiting superior robustness, scalability, and potential for real-time applications.

This work addresses the challenge of accurately modeling viscous flow in highly porous media using physics-informed neural networks (PINNs), where complex boundaries and fine-scale structures lead to significant accuracy degradation—particularly as pore count increases. To overcome this, the authors propose a novel approach that integrates finite basis PINNs (FBPINNs) with hard boundary constraints. By decomposing the domain into local subregions, the method precisely embeds pore boundary conditions, effectively mitigating spectral bias and non-local effects. Notably, this is the first application of hard constraints within FBPINNs for porous flow problems, circumventing the stiffness and gradient conflicts associated with soft constraints and yielding convergence behavior that is largely independent of pore number. Numerical experiments demonstrate that the proposed framework achieves high accuracy, computational efficiency, and strong scalability in simulating Stokes flow.

Neural networks often struggle to strictly satisfy nonlinear constraints during inference, which hinders their deployment in safety-critical applications. This work proposes HardNet++, the first method capable of enforcing hard satisfaction of general nonlinear equality and inequality constraints, overcoming the limitation of existing approaches that are restricted to specific constraint forms. By integrating damped local linearization, differentiable projection layers, and end-to-end training, HardNet++ guarantees constraint compliance simultaneously during both training and inference. Evaluated on model predictive control tasks, HardNet++ achieves high-precision constraint adherence while preserving solution optimality.