asymptotic-preserving analysis

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.

To address the insufficient time-integration capabilities of the SUNDIALS numerical library in high-performance scientific computing, this project systematically extends its time-stepping solvers. Methodologically, it introduces three novel classes of single-step methods—low-storage Runge–Kutta (LSRK), symplectic structure-preserving block RK, and general-purpose operator-splitting schemes—alongside a new multi-rate adaptive step-size controller and explicit RK-based adjoint sensitivity analysis (filling a longstanding gap). It further enhances nonlinear solvers with Anderson acceleration and improves error handling and logging infrastructure. These contributions significantly improve efficiency, stability, and accuracy for large-scale transient simulations, enabling long-duration, high-fidelity, and multiphysics-coupled modeling. Validation across multiple HPC applications demonstrates speedups of 1.5–3× and markedly improved numerical robustness.

Physics-informed neural networks (PINNs) suffer from poor linear stability when solving stiff initial-value problems of ordinary differential equations (ODEs). Method: We propose physics-informed random projection neural networks (PI-RPNNs), introducing a novel random projection framework with both multi-point and single-point collocation strategies. Contribution/Results: We provide the first rigorous proof that multi-point PI-RPNNs achieve asymptotic stability for highly stiff linear ODE systems, while single-point PI-RPNNs satisfy A-stability. This establishes the first unified convergence and stability theory for RP-PINNs applied to stiff linear ODEs and parabolic partial differential equations (PDEs). Compared with classical implicit methods—including backward Euler, Crank–Nicolson, and Radau schemes—PI-RPNNs deliver significantly higher numerical accuracy and computational efficiency over wide step-size ranges. The method overcomes the longstanding trade-off between stability and accuracy in traditional numerical solvers, providing both theoretical foundations and practical tools for reliable PINN-based simulation of stiff problems.

This work addresses the convergence guarantees of stochastic line search optimization for over-parameterized models under interpolation conditions. We establish a necessary and sufficient condition on the search direction—applicable to a broad class of methods—that ensures finite termination and bounded backtracking steps, and rigorously prove linear convergence under the Polyak–Łojasiewicz (PL) assumption. The condition unifies major first-order strategies—including momentum, conjugate gradient, and adaptive preconditioning—providing a verifiable theoretical foundation for their principled integration with stochastic line search. Our analysis fills a critical gap in the convergence theory of stochastic line search methods and significantly extends both the applicability and reliability of efficient first-order optimization in interpolation learning regimes.

This work identifies the fundamental limitations of shallow neural networks in approximating and learning high-frequency signals under finite machine precision and computational constraints. Methodologically, it employs matrix condition number analysis, gradient flow modeling, frequency-domain error decomposition, and large-scale numerical experiments. The study establishes, for the first time, quantitative lower bounds on numerical approximation error, characterizes computational complexity bottlenecks, and reveals condition-number-driven instability in high-frequency approximation—thereby unifying accuracy, cost, and stability into a precise theoretical framework. Theoretical analysis and empirical validation jointly demonstrate that: (i) approximation error for high-frequency components is dominated by exponentially deteriorating condition numbers; (ii) the error lower bound grows exponentially with signal frequency; and (iii) computational cost increases polynomially to exponentially with frequency. These results provide critical theoretical criteria and practical design principles for modeling high-frequency signals with neural networks.

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 proposes a stabilization framework based on score-based generative models to address non-physical instabilities and structural distortions commonly encountered in the numerical solution of time-dependent partial differential equations (PDEs). For the first time, score-based generative models are introduced into PDE numerical stabilization, where a conditional stabilizing operator with manifold-contracting properties is constructed by learning the physically admissible solution manifold. This operator corrects intermediate solutions during time integration. Numerical experiments demonstrate that the method significantly enhances robustness for convection, Korteweg–de Vries (KdV), nonlinear Schrödinger, and Burgers equations, effectively suppressing spurious oscillations while preserving essential dynamical features.

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.

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 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.