boundary condition handling

Numerical solvers for nonlinear hyperbolic PDEs often suffer from spurious shocks, instability, violation of conservation laws, or convergence to nonphysical solutions. Method: We propose the first end-to-end formal verification framework for such solvers, built on Racket metaprogramming and a custom theorem prover. It integrates symbolic automatic differentiation with floating-point algebra-aware modeling, enabling user-defined physical models and automatic generation of mathematically verified C code. Contribution/Results: The framework formally verifies L² stability, flux conservation, and physical validity, and has been integrated into the Gkeyll multiphysics platform. Experiments demonstrate that the generated solvers achieve both high performance and physical fidelity in strongly nonlinear regimes—including shock waves and turbulence—while providing, for the first time, full formal correctness guarantees from high-order algorithm specification to executable code.

This work addresses the challenges of complex boundary condition handling, code redundancy, and poor distributed efficiency in partial differential equation solvers on block-structured grids. The authors propose a unified modeling approach that expresses user-defined boundary conditions as affine sparse linear operators and, for the first time, systematically reformulates them into sparse matrix-vector multiplication (SpMV) form. Leveraging a domain-specific language (DSL) and compiler techniques—combined with multi-stage programming and polyhedral analysis—the framework automatically generates highly optimized matrix-free or sparse matrix kernels while optimizing communication scheduling and reuse. The method achieves substantial performance gains, demonstrating 72%–88% strong scaling efficiency on 1,344 CPU cores, up to 7.6× acceleration in boundary computation kernels, and a reduction of over 70% in code size.

This work addresses the instability arising when enforcing Neumann and Robin boundary conditions on piecewise $C^1$ (globally only $C^0$) boundaries in Physics-Informed Neural Operators (PINOs). We propose a novel strong-constraint method based on orthogonal projection, extending the Sukumar & Srivastava framework to relax the conventional requirement of globally $C^1$ boundaries. By constructing admissible trial functions and applying orthogonal projection onto the constrained function space, our approach ensures exact and numerically stable satisfaction of boundary conditions. The method unifies weak, semi-weak, and strong formulations for boundary treatment. We validate it on scalar Darcy flow and steady-state Navier–Stokes equations: results demonstrate significantly improved training stability and numerical accuracy, enhanced robustness on complex geometries, faster convergence, and lower approximation errors compared to standard approaches.

This work addresses the frequent mismatch between user-specified physical intent and the actual behavior of multiphysics simulation code generated by large language models, often due to erroneous implementations of partial differential equations (PDEs). To bridge this gap, we propose a PDE-structure-based intent verification method that deterministically reconstructs the governing equations implicitly encoded in the generated code and compares them against the user’s intended PDEs, enabling semantic correctness validation and iterative refinement. We introduce, for the first time, a formal metric termed the Intent Fidelity Score (IFS) to quantify alignment with physical intent, establish a PDE-driven feedback loop, and demonstrate compatibility with major PDE frameworks including MOOSE, FEniCS, and FreeFEM. Evaluated on 220 cases in MooseBench, our approach substantially improves IFS—by 0.22–0.41 on challenging instances with initial IFS < 0.7—while audits reveal that execution-only repair strategies still yield physically incorrect results in 39–40% of cases.

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 a key limitation of conventional LLM-based PDE solvers, which implicitly embed numerical strategies within generated code, making pre-execution validation and post-failure correction challenging. To overcome this, the authors propose AutoPDE, the first framework to explicitly model solution strategies as revisable, decoupled objects separate from implementation code. AutoPDE employs a three-stage pipeline—PDE type identification, numerical method selection, and adaptive parameter tuning—augmented by low-overhead trial solves and a reusable skill library to construct and refine strategies prior to code generation. Evaluated on the PDE Agent Bench, AutoPDE achieves a 54.5% pass rate, outperforming the strongest baseline by 14.2 percentage points, thereby substantially improving both the reliability and interpretability of AI-driven PDE solving.

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.

This work addresses the behavioral gap between formal verification and actual execution in traditional engineering approaches, which often neglect execution semantics. To bridge this semantic divide, the paper proposes a Modeling and Simulation-Based Engineering (MSBE) methodology that explicitly treats execution semantics as a first-class engineering entity. It defines executability as the admissible model space induced by the stabilization of execution conditions and unifies model behavior with physical execution through an iterative cycle of formal execution, experimental execution, verification, and activity-mediated validation. Integrating formal methods, simulation-based verification, activity theory, and constraint modeling, MSBE establishes a general-purpose engineering framework applicable to diverse cyber-physical systems (CPS). The approach demonstrates its generality and effectiveness across four CPS categories: human-centric, biophysical, technological, and digital twin systems.

This work addresses the challenge of formal verification for Reflex programs in industrial-scale control systems, where the generation of an excessive number of verification conditions often renders manual analysis impractical. To overcome this limitation, the authors propose a hybrid verification strategy that integrates a structured requirement annotation language with automated invariant inference based on program structure, coupled with an SMT solver to automatically discharge a substantial subset of verification conditions. By leveraging this synergistic approach, the method significantly reduces the number of verification tasks requiring human intervention, thereby enhancing the automation, feasibility, and overall efficiency of formal verification for large-scale process control systems.

This work addresses the limitations of traditional PDE solvers, which rely heavily on expert knowledge and laborious development, as well as existing large language model (LLM) approaches that focus primarily on reasoning optimization while lacking fine-grained feedback on scientific computation accuracy. The authors propose RLVP, a novel framework that introduces, for the first time, a physics-consistency-based continuous reward mechanism combined with a hard constraint on program executability to train LLMs via reinforcement learning for generating high-accuracy solver code. This approach overcomes the shortcomings of conventional binary verification in scientific computing, significantly outperforming both pretrained and supervised fine-tuning baselines across multiple PDE benchmarks. Notably, even smaller models trained with RLVP surpass state-of-the-art prompting strategies of larger models and demonstrate strong zero-shot transfer across PDE types and compositional generalization of numerical modules.