discretization and spatial discretization methods

This work addresses a key limitation of conventional neural PDE surrogates, which model field evolution on fixed grids and thereby overlook the critical role of mesh design in allocating spatial resolution and spectral bandwidth. The study introduces, for the first time, adaptive discretization as a physics-constrained conditional generation task, proposing a two-stage diffusion framework: it first generates an r-adaptive displacement grid conditioned on observed dynamics and then predicts solution evolution on this adaptive mesh. By incorporating physics-aware regularization, geometric validity constraints, and local spectral concentration, the method achieves learnable, interpretable, and numerically stable mesh adaptation. Extensive experiments across five classes of PDE problems demonstrate substantial improvements over traditional adaptive and reduced-order methods, with particularly notable gains in complex domains.

To address the computational inefficiency and trade-off between accuracy and efficiency in traditional adaptive finite element method (FEM) mesh generation for solving partial differential equations (PDEs) on complex 3D geometries—such as turbine blade scans—this paper proposes an end-to-end learning-based adaptive meshing framework. Our method employs a lightweight neural network to directly regress a spatial size field from sparse Monte Carlo (MC) solution estimates, replacing iterative optimization with a single forward inference pass. The pipeline integrates Monte Carlo sampling, neural regression, size-field-driven tetrahedral mesh generation, and FEM solving. Evaluated across diverse 3D shapes and boundary conditions, our approach achieves 2–4× speedup over conventional adaptive FEM and MC-based methods while maintaining comparable solution accuracy and demonstrating strong generalization capability.

This work addresses the challenge that traditional neural operators struggle with discontinuities in discrete structures arising from topological changes, abrupt boundary variations, or shifts in computational domains. To overcome this limitation, the authors propose a novel paradigm termed Discrete Solution Operator Learning (DiSOL), which decomposes the solution process into three learnable stages: local encoding, multiscale assembly, and implicit reconstruction. Unlike conventional approaches that rely on continuous function mappings and smooth geometric variations, DiSOL emphasizes process-level consistency. By integrating classical discretization principles with deep learning, the method encodes local contributions on embedded meshes and aggregates information across multiple scales. It demonstrates high-accuracy and robust predictions—both in-distribution and under strong out-of-distribution scenarios—for problems involving complex geometric changes, including Poisson equations, convection-diffusion, linear elasticity, and heat conduction.

This work addresses the high memory and computational complexity typically associated with solving three-dimensional partial differential equations on Cartesian grids. By exploiting tensor-product structure, the proposed method decomposes the 3D operator into one-dimensional banded kernels aligned with coordinate axes, thereby avoiding explicit assembly of the global matrix and enabling a matrix-free solution strategy. Within a unified framework that integrates diverse numerical approaches—including Kronecker product algebra, compact finite differences, isogeometric analysis, and direct diagonalization—the study systematically identifies three key techniques: multi-right-hand-side reshaping, sum factorization, and pencil-style MPI decomposition. These innovations collectively enhance hardware affinity and parallel scalability, reducing algorithmic complexity to O(N) and storage requirements to O(Nₓ + Nᵧ + N_z), thus enabling efficient large-scale 3D PDE simulations.

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.

Existing neural PDE solvers lack rigorous theoretical guarantees linking residual errors to solution-space errors, making their generalization performance difficult to quantify. This work establishes a unified theoretical framework that, under the assumption of a compact solution subset, leverages functional space analysis, generalization theory, and probabilistic inequalities to derive, for the first time, explicit and certifiable deterministic and probabilistic generalization bounds relating pointwise collocation residuals, initial condition errors, and boundary condition errors to the overall solution error. By bridging this gap, the study fills a critical theoretical void in physics-informed neural networks regarding solution error control and provides rigorous convergence and reliability guarantees for residual-based training methodologies.

This work addresses the limitations of traditional numerical methods—prohibitive computational cost in high-dimensional and geometrically complex settings—and the lack of theoretical guarantees and poor generalization in current machine learning approaches for solving partial differential equations (PDEs). To bridge this gap, the authors propose a hybrid PDE-solving paradigm that integrates deductive numerical schemes with inductive learning models. They establish a unified evaluation framework encompassing six core computational challenges and, from an epistemological perspective, formally distinguish between the two methodological classes. By introducing a structure inheritance mechanism and an error budget decomposition, they clarify the conditions under which theoretical guarantees propagate through the hybrid system. Leveraging physics-informed neural networks, differentiable programming, foundation models, and quantum algorithms, the study constructs a multi-paradigm collaborative framework and articulates responsible criteria for method selection, revealing three forms of complementarity that enable scalable, theoretically grounded simulation of high-dimensional complex systems.

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 inefficiency of traditional finite difference methods on uniform grids in resolving localized complex features—such as shocks or steep gradients—without excessive degrees of freedom. The authors propose a hybrid adaptive strategy that leverages the residual of a physics-informed neural network (PINN) as an off-grid indicator to generate cell-wise refinement metrics. These metrics drive non-uniform mesh refinement through a combination of Dörfler and threshold-based marking criteria, while high-fidelity solutions are still computed using finite differences. Innovatively, the PINN serves not as a solver but as an adaptive guidance tool, effectively integrating physical priors with classical numerical methods. In one-dimensional viscous Burgers’ equation, the approach achieves superior accuracy with only 60 degrees of freedom—outperforming a uniform 192-point grid by reducing error by 67.5%—and demonstrates marked advantages over random refinement strategies in two- and three-dimensional proxy problems.

This work addresses the dominant role of discretization error in reverse-time sampling of diffusion models under a fixed inference budget, a factor overlooked by existing non-asymptotic analyses that are often loose and ignore data structure. By deriving first-order asymptotic expansions for both the weak error of the Euler–Maruyama scheme and the Fréchet discretization error, the paper establishes—for the first time—an explicit connection between this error and intrinsic data geometry, such as the covariance spectrum, as well as the diffusion schedule. Under the exact score assumption, the integration of numerical analysis for stochastic differential equations with asymptotic theory yields a computable, geometry-aware optimization objective in the Gaussian setting. This formulation demonstrates strong predictive performance across diverse image generation and posterior sampling tasks, offering both theoretical grounding and practical tools for geometry-informed schedule design.