analytical solution comparison

This work addresses the absence of a standardized benchmark for evaluating code generation targeting partial differential equation (PDE) solvers, particularly with respect to numerical accuracy, computational efficiency, and compatibility with mainstream finite element libraries. To bridge this gap, the authors introduce the first multi-metric, multi-library benchmark for PDE solver generation, comprising 645 structured instances spanning six mathematical problem types and eleven PDE classes. The benchmark supports three major finite element frameworks—DOLFINx, Firedrake, and deal.II—and incorporates a staged evaluation framework that holistically assesses code executability, numerical correctness, and performance. Experimental results demonstrate that while current large language models can produce executable code, their success rate drops substantially when stringent accuracy and efficiency requirements are imposed, thereby underscoring the necessity and effectiveness of the proposed benchmark in advancing reliable and efficient automated PDE solver generation.

Existing optimization algorithm benchmarks often lack engineering realism, failing to reflect practical performance. Method: We construct a highly diverse benchmark suite comprising 231 continuous, unconstrained engineering optimization problems derived from real-world CFD and FEA modeling. We further propose, for the first time, a nonlinear normalization technique for performance evaluation, using random sampling as a statistical baseline—enabling unbiased, reproducible efficiency comparisons across heterogeneous problems. Contribution/Results: Based on hundreds of independent algorithm runs, we robustly evaluate 20 deterministic and stochastic optimizers. Results reveal that most widely used metaheuristics exhibit significantly suboptimal efficiency on engineering problems, while only a few demonstrate superior performance. This benchmark framework enhances the authenticity, transparency, and practical relevance of algorithm assessment, establishing a new paradigm for selecting and improving optimization algorithms tailored to engineering applications.

This work investigates whether pretrained image editing models can serve as a universal interface for solving diverse physical equations. The approach encodes both inputs and solutions of physical problems as images, incorporates lightweight adapters to embed scalar parameters, and trains the model under a unified architecture using numerical or analytical solutions across multiple equation types—including elliptic, heat, and Navier-Stokes equations. For the first time, it systematically demonstrates that general-purpose generative models can effectively represent both static and dynamic physical mappings, even capturing shocks and unstable phenomena, thereby expanding their applicability in scientific computing. Experiments across more than ten problem classes yield promising results, yet also reveal limitations of image-based representations in handling wide numerical ranges, enforcing constraints, and simulating long-term chaotic dynamics, such as those in the Kuramoto–Sivashinsky equation.

Current LLM evaluations lack scientifically valid benchmarks tailored to computational mechanics, hindering rigorous assessment of their ability to generate physically consistent and numerically reliable finite element method (FEM) code. To address this, we propose FEM-Bench—the first structured scientific reasoning benchmark that systematically integrates physical principles (e.g., partial differential equations, constitutive relations) with numerical constraints (e.g., mesh discretization, element assembly), covering core modeling tasks including geometric representation, material behavior, and boundary condition specification. We introduce a novel “physics–mathematics” dual-constraint criterion for verifiable scientific code generation, enabling progressive difficulty scaling and objective quantification. Coupled with an automated unit-testing framework, FEM-Bench evaluates 33 nontrivial introductory tasks: Gemini 3 Pro achieves a 78.8% full-pass rate in function generation, while GPT-5 attains a 73.8% pass rate in joint testing—revealing critical limitations of current LLMs in rigorous scientific programming.

Existing machine learning solvers—such as PINNs, FNOs, and DeepONets—excel on single-field PDEs but lack systematic evaluation and methodological adaptation for strongly coupled multiphysics systems governed by multi-physics PDEs. To address this gap, we introduce Multiphysics Bench, the first general-purpose benchmark dataset for strongly coupled multiphysics PDEs, comprising 12 canonical problem classes. Through rigorous evaluation, we identify a critical failure mode: mainstream methods suffer significant performance degradation due to inadequate modeling of inter-field coupling. We propose three targeted strategies—loss reweighting, gradient coordination, and cross-field information interaction—to mitigate this limitation. Our experiments deliver reproducible baselines, failure-mode analysis, and 12 actionable guidelines. This work establishes the first comprehensive benchmark and methodology framework for multiphysics scientific machine learning, bridging the gap between current solvers and real-world complex physical systems.

Existing differentiable PDE solvers lack a unified benchmark to evaluate their practical performance in terms of gradient correctness, computational overhead, numerical stability, and ease of integration. This work proposes an extensible benchmarking framework that, through containerized encapsulation (Tesseract) and a standardized gradient API, supports cross-language interoperability and multiple automatic differentiation strategies. For the first time, it enables a systematic comparison of 14 differentiable solvers spanning fluid dynamics, structural mechanics, and heat transfer. Experiments reveal order-of-magnitude differences in computational cost and Jacobian condition numbers across solvers, yet all converge to similar optimal solutions. These findings indicate that real-world bottlenecks lie primarily in memory consumption, numerical stability, and compatibility—not in optimization capability.

This work addresses the interoperability challenges arising from inconsistent interfaces among numerical solvers by proposing and implementing MaRDI—a standardized, open interface tailored for nonlinear optimization. Designed with a modular architecture, MaRDI establishes a generic solver adapter layer that enables seamless integration of diverse optimizers and embeds naturally within physics-informed neural network (PINN) training pipelines. Its efficacy is demonstrated through application to the viscous Burgers equation, where it substantially reduces the development overhead and benchmarking costs associated with solver-specific bindings. By abstracting low-level implementation details, MaRDI allows researchers to focus on core algorithmic innovation while significantly enhancing the efficiency and reproducibility of cross-solver experimentation.

This study investigates whether the internal mechanisms of the scientific foundation model Walrus align with physical principles when reproducing continuum dynamics, and examines the relationship between its representations and performance. By introducing sparse autoencoders (SAEs) at specific layers, the work pioneers the use of enstrophy—the integral of squared vorticity—for physically grounded filtering and prioritization of large-scale features, complemented by comparative numerical simulations. The findings reveal that while the model’s feature activations exhibit segment-wise consistency, they do not correspond to physically meaningful decompositions. Notably, certain output inaccuracies, such as excessive energy dissipation, can be traced to variations in specific SAE features. The study underscores fundamental challenges in achieving representational fidelity and interpretability in scientific foundation models.

This work addresses the challenge that existing frameworks struggle to efficiently support the direct development of multiphysics coupling solvers and parallel adaptive simulations. We propose a portable, reproducible cross-language framework that, for the first time, tightly integrates Trixi.jl with deal.II to construct a strongly coupled partitioned solver for Newtonian self-gravitating hydrodynamics. By combining a strong coupling strategy, cross-language interoperability, adaptive mesh refinement, and parallel computing, our framework substantially simplifies the development of coupled solvers. Numerical experiments demonstrate high-order convergence, physical consistency, and effective adaptivity of the algorithm, while also exhibiting favorable strong scaling performance in parallel execution.

This work addresses the inefficiency of traditional harmonic balance methods in analyzing periodic responses of high-dimensional, complex nonlinear systems, which typically require manual provision of partial derivatives of nonlinear terms. To overcome this limitation, the authors propose an open-source, modular semi-analytical framework that integrates PyTorch’s automatic differentiation with the harmonic balance method. The framework enables fully automated end-to-end computation from user-specified equations of motion without manual derivation. It incorporates weighted arc-length continuation, sparse matrix assembly, block-wise solvers, and Floquet stability analysis, significantly enhancing scalability and computational efficiency. Demonstrated on a system with 2,000 degrees of freedom (202,000 unknowns), the method achieves a single-point solution in just 0.44 seconds—hundreds of times faster than the Newmark-β method—while maintaining manageable memory usage.