apply finite-difference methods

This paper addresses the challenge of coordinating geographically distributed, heterogeneous, and autonomous computing resources. Method: It systematically introduces the “computational grid” concept and architecture for building a scalable, secure, and transparent virtual supercomputer. Key innovations include resource virtualization, cross-domain trust mechanisms, a unified naming and scheduling model, a distributed middleware framework, resource discovery and scheduling algorithms, a prototype security authentication protocol based on the Grid Security Infrastructure (GSI), and a cross-platform communication standard. Contribution/Results: The work establishes the foundational paradigm of grid computing, providing both theoretical grounding and practical implementation blueprints. It directly enabled the development of core infrastructure—including the Globus Toolkit—and catalyzed the advancement of e-Science. Moreover, it served as a seminal intellectual precursor to modern cloud and edge computing paradigms.

This work addresses the longstanding challenge in meshfree methods of simultaneously achieving high accuracy, computational efficiency, and robustness to irregular point distributions when approximating differential operators. The authors propose a self-supervised graph neural network that directly learns discrete differential operator weights from local point-set geometry, incorporating polynomial moment constraints derived from truncated Taylor expansions to enforce consistency. This approach yields, for the first time, a high-accuracy operator that depends solely on local geometry, is resolution-independent, and generalizes across diverse point configurations—combining the classical polynomial reproduction property with robustness to arbitrary node layouts. Experiments demonstrate superior accuracy over standard SPH on benchmark problems, a more favorable accuracy–efficiency trade-off than high-order consistent meshfree methods at moderate precision levels, and successful application to solving weakly compressible Navier–Stokes equations.

To address the demand for large-scale PDE simulations in seismic and medical imaging, this paper proposes a fully automated high-performance code generation method tailored to explicit finite-difference (FD) stencils. The approach deeply integrates MPI-X (including UCX and shared memory) distributed parallel code generation into the Devito domain-specific language (DSL) compilation pipeline—enabling end-to-end automation from symbolic modeling to HPC-ready code without source-code modifications, and supporting scalable CPU/GPU cross-platform execution. Key techniques include symbolic differentiation, loop optimization, communication–computation overlap, and GPU offloading. Experiments on multi-node CPU/GPU clusters demonstrate excellent strong and weak scaling, substantial reduction in execution time, and over 70% decrease in developer effort for coding and performance tuning. The framework has been successfully deployed in production-scale scientific computing tasks, including real-world seismic full-waveform inversion.

Neural operator training is severely constrained by reliance on data generated via traditional numerical solvers, which are computationally expensive, discretization-dependent, and introduce approximation errors. Method: We propose a solver-free backward synthetic data generation framework: candidate solutions $u_j$ are randomly sampled from the solution space (e.g., $H_0^1(Omega)$), and their exact source terms $f_j = mathcal{L}u_j$ are computed directly via automatic differentiation—yielding infinite, zero-error $(f_j, u_j)$ training pairs without numerical discretization. Contribution/Results: This work introduces the first “solution-to-source” inverse generation paradigm, eliminating dependence on numerical solvers while preserving mathematical rigor and computational scalability. Experiments demonstrate that neural operators trained exclusively on synthetic data achieve generalization performance on multiple PDE benchmarks comparable to—or even exceeding—that of models trained on solver-generated data.

Sensitivity analysis for steady-state heat conduction in heterogeneous materials—characterized by strong phase contrast and temperature-dependent properties—is computationally expensive when performed via conventional adjoint methods. Method: This paper proposes the Finite Operator Learning (FOL) framework, which tightly integrates neural operators with finite element discretization. FOL embeds physical constraints—including the weak-form energy functional, boundary conditions, and residual stationarity—into a multi-objective loss function, and combines Sobolev-norm training with feedforward networks to jointly predict both PDE solutions and their sensitivities to design parameters in an end-to-end manner. Contribution/Results: FOL requires neither labeled training data nor adjoint computations, ensuring strong physics consistency. It directly outputs high-fidelity solutions and accurate gradients, enabling tangent-matrix-driven microstructural thermal optimization. By eliminating iterative adjoint solves, FOL significantly reduces sensitivity analysis cost while preserving numerical robustness and physical fidelity.

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.

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 the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work addresses the unclear relationship between continuous theory and discrete implementation in neural operators for solving partial differential equations, particularly concerning stability and discretization error. For the first time, rigorous discretization error bounds are established for State-Space Neural Operators (SS-NOs) and Fourier Neural Operators (FNOs). By leveraging functional analysis, the regularity of solutions is explicitly linked to input discretization, and Input-to-State Stability (ISS) theory is introduced to quantify how discretization affects stability in the continuous domain. Numerical experiments on one- and two-dimensional benchmark problems validate the tightness of the derived theoretical bounds, demonstrating that SS-NOs exhibit both robustness and numerical stability across varying resolutions.

This work addresses the high memory overhead and neglect of local structure in gradient computation for implicit nonlinear solvers within differentiable simulation. The authors propose a solver-level differentiation method that constructs an adjoint algorithm symmetric to the forward solve by reverse-scanning a block-structured implicit solver, entirely avoiding the assembly of a global Jacobian matrix. For the first time, adjoint computation is aligned with the block structure of the forward solver, combining vertex-block descent with reverse-colored Gauss–Seidel sweeps to enable efficient backpropagation using only local 3×3 adjoint solves. This approach leverages operator-view approximations of the inverse and its transpose. On a single GPU, it achieves a 33× speedup and 71× reduction in memory compared to unrolled automatic differentiation, enabling, for the first time, differentiable elastic dynamics simulation of million-contact coupled soft bodies with up to 8 million vertices.