ode/pde solver implementation and interfacing

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 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 study systematically compares MATLAB, Mathematica, and Maple in solving ordinary differential equations (ODEs), partial differential equations (PDEs), and systems of differential equations. A unified benchmark suite—grounded in analytically tractable reference solutions—is employed to empirically evaluate the tools across five dimensions: syntactic usability, numerical accuracy, computational efficiency, visualization capability, and specialized solver functionality. Crucially, the work introduces a novel, problem-driven software selection framework that classifies tasks by intrinsic characteristics—including equation type, stiffness, and boundary condition complexity. Results indicate that Mathematica excels in symbolic solution derivation and medium-scale ODE accuracy; MATLAB demonstrates superior performance in large-scale numerical simulation and engineering-oriented PDE applications; and Maple offers distinctive advantages in special-function handling and analytic derivation. This is the first systematic, multidimensional comparative study of these major mathematical software platforms, thereby bridging a critical gap in computational tool evaluation and providing actionable, evidence-based guidance for scientific and engineering practice.

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 PDE solvers rely heavily on manual modeling and domain expertise, while physics-informed neural networks (PINNs) lack autonomy and natural language interaction capabilities. Method: We propose the first end-to-end, natural-language-driven PDE solving system. Its core innovations include: (1) a tool-augmented multi-agent collaboration framework integrating large language model (LLM) reasoning with controllable, deterministic tool invocation; (2) Prog-Act graph memory—a dual-loop dynamic planning mechanism with built-in error correction, coupled with a decoupled architecture separating resource pools from tool parameters to resolve multi-tool dependency and cross-step coordination challenges; and (3) the open-source PDE-Bench benchmark and hierarchical evaluation metrics. Results: Experiments demonstrate significant performance gains over state-of-the-art methods on complex, multi-step PDE tasks. The codebase and dataset are publicly released to advance automated scientific computing.

This work proposes an end-to-end scientific workflow framework for partial differential equations (PDEs) based on large language models (LLMs), aiming to bridge the gap between simulation and real-world systems. The framework systematically integrates LLMs across the entire PDE pipeline—spanning discovery of governing equations, automated generation and iterative refinement of numerical solvers, and simulation-informed decision-making—thereby establishing an intelligent interface that connects natural language, symbolic mathematics, executable code, and physical constraints. Experimental results demonstrate the framework’s significant potential in automating PDE-centric scientific workflows, while also highlighting critical challenges such as the scarcity of high-quality data and difficulties in transferring learned capabilities to real-world scenarios.

This work addresses the challenge that effective neural PDE solvers are extremely sparse in the design space, rendering large language models inefficient for their automated discovery. To overcome this, the authors propose ADSL-PDE, a framework that introduces structured search states to decouple high-level design decisions—such as architecture, physical constraints, and optimization objectives—from low-level code implementation. By formulating a domain-specific language to restructure the search space, the method substantially increases the density of valid candidate solvers. Building upon this representation, the framework integrates deterministic compiler-based mapping with an evolutionary algorithm guided by empirical feedback and large language model suggestions. Experiments across multiple PDE benchmarks demonstrate that the approach improves solver performance by over 52% within the first ten iterations, while significantly enhancing search efficiency and optimization stability.

This work proposes the first end-to-end autonomous agent framework that automatically translates user-specified ordinary and partial differential equation (ODE/PDE) problems—expressed in natural language—into trainable and inferable neural operators. The approach employs a stateful input graph to parse semantic intent, invokes the FEniCSx finite element solver to generate high-fidelity training data, and trains a multi-branch Bayesian DeepONet architecture to learn the underlying solution operator. The system supports interactive multi-turn editing, parameterized data generation, and modular deployment, substantially enhancing automation and reproducibility in scientific machine learning workflows. Evaluated across multiple ODE/PDE benchmarks, the framework achieves efficient, solver-free predictions without reliance on traditional numerical solvers.

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.

This work addresses the lack of efficient, end-to-end differentiable solvers for differential-algebraic equations (DAEs) arising from multiphysics systems, which hinders parameter inversion and optimal control. We present the first natively JAX-based differentiable DAE solver that unifies forward simulation with reverse-mode automatic differentiation. The solver integrates adaptive BDF, Radau, and Rosenbrock methods, combined with Pantelides index reduction and dummy derivative techniques. By freezing the forward time-step grid and re-solving a variable-step BDF-2 scheme on this fixed mesh, gradient propagation becomes highly efficient, enabling DAEs to serve as differentiable primitives. A single call to jax.grad computes full gradients, and wall-clock time remains nearly constant when scaling batched parameter sweeps from 1 to 1,000, dramatically improving differentiable computation efficiency for multiphysics DAE systems.