生成时间: 2026年05月11日 12:56:31

重点关注论文数量: 0

了解领域论文数量: 1

不相关论文数量: 23

了解领域论文（摘要翻译）

1. 关于双曲型方程组的保结构与逐点守恒连续-间断伽辽金格式

On structure-preserving and pointwise conservative continuous DG schemes for hyperbolic systems

作者: Rémi Abgrall, Michael Dumbser, Pierre-Henri Maire, Enrico Zampa

类别: math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07576v1

链接: http://arxiv.org/abs/2605.07576v1

摘要翻译

中文标题: 关于双曲型方程组的保结构与逐点守恒连续-间断伽辽金格式

摘要翻译: 我们提出一类新的保结构半离散连续-间断伽辽金(CG-DG)有限元格式，用于非结构单纯形网格上的线性和非线性双曲型偏微分方程组，其自动满足以下性质：i) 新格式不仅是单元逐块守恒的，而且在每一点都是局部逐点守恒的，因此它们在任意控制体积上满足守恒律的积分形式，而这些控制体积完全不必与网格重合；ii) 新方法在离散层次上自然而精确地逐点局部和全局满足两个基本的向量分析恒等式 $\nabla \cdot \nabla \times \mathbf{A}$ 和 $\nabla \times \nabla Z$；iii) 对于线性对称双曲型系统，格式关于平方能量是自然能量守恒的，即在 $L^2$ 范数下是非线性稳定的。新型CG-DG格式的关键在于使用了两个不同但相容的逼近空间：由次数不超过 $N$ 的分片不连续多项式构成的经典DG空间 $\mathcal{U}_h^N$，和由次数为 $N+1$ 的全局连续分片多项式构成的经典有限元空间 $\mathcal{W}_h^{N+1}$。在新的CG-DG格式中，离散解 $\mathbf{u}_h$ 在 $\mathcal{U}_h^N$ 中寻求，而合适的离散通量场 $\tilde{\mathbf{f}}_h$ 在 $\mathcal{W}_h^{N+1}$ 中计算。对于 $N=0$，我们的新格式直接与具有适当顶点基通量的格心有限体积格式相关。该格式的所有声称性质均首先通过数学证明，随后经由适当的数值试验加以验证。我们展示了我们的方法在三个线性和非线性双曲型系统上的应用。

不相关论文（基本信息）

1. 守恒律方程组时间显式与时间隐式离散的不变域保持限制技术

Invariant domain preserving limiting of time explicit and time implicit discretizations for systems of conservation laws

作者: Bartolomeo Fanizza, Florent Renac

类别: math.NA

发布日期: 2026-05-08

arXiv ID: 2605.08038v1

链接: http://arxiv.org/abs/2605.08038v1

摘要: This work concerns the design and analysis of a limiting technique that allows the preservation of invariant domains for high-order numerical approximations of nonlinear hyperbolic systems of conservation laws. The method can be applied to any conservative discretization method in space as well as to a wide range of explicit and implicit time integration schemes. The method limits the high-order solution around a low-order accurate solution that is known to preserve all the invariant domains. It generalizes the flux-corrected transport limiter [J. P. Boris and D. L. Book, J. Comput. Phys., 11, 1973; S. T. Zalesak, J. Comput. Phys., 31, 1979] to systems of conservation laws and relies on the limitation of antidiffusive fluxes, but defines the limiting coefficients so as to express the limited solution as a convex combination of invariant domain preserving quantities similarly to the convex limiting framework [Guermond et al., Comput. Methods Appl. Mech. Engrg., 347, 2019]. We give details on the derivation of this limiting technique and provide some illustration with finite volume or discontinuous Galerkin (DG) space discretizations associated to explicit or implicit Runge-Kutta methods as well as to time DG integrations. The limiter is applied iteratively to refine the limited solution around the high-order one, while preserving the invariant domains, and a heuristic is proposed to accelerate its convergence. Numerical experiments solving one- and two-dimensional problems involving scalar hyperbolic equations and the compressible Euler equations are presented to illustrate the properties of these schemes.

2. 利用高效随机龙格-库塔方法加速郎之万蒙特卡洛：超越对数凹性

Accelerating Langevin Monte Carlo via Efficient Stochastic Runge–Kutta Methods beyond Log-Concavity

作者: Bin Yang, Xiaojie Wang

类别: math.ST, math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07939v1

链接: http://arxiv.org/abs/2605.07939v1

摘要: Sampling from a high-dimensional probability distribution is a fundamental algorithmic task arising in wide-ranging applications across multiple disciplines, including scientific computing, computational statistics and machine learning. Langevin Monte Carlo (LMC) algorithms are among the most widely used sampling methods in high-dimensional settings. This paper introduces a novel higher-order and Hessian-free LMC sampling algorithm based on an efficient stochastic Runge–Kutta method of strong order $1.5$ for the overdamped Langevin dynamics. In contrast to the existing Runge–Kutta type LMC (Li et al., 2019) involved with three gradient evaluations, the newly proposed algorithm is computationally cheaper and requires only two gradient evaluations for one iteration. Under certain log-smooth conditions, non-asymptotic error bounds of the proposed algorithms are analyzed in $\mathcal{W}_2$-distance. In particular, a uniform-in-time convergence rate of order $O(d ^{\frac32} h^{\frac32})$ is derived in a non-log-concave setting, matching the convergence rate proved in the aforementioned work but under the log-concavity condition. Numerical experiments are finally presented to demonstrate the effectiveness of the new sampling algorithm.

3. NSPOD：通过近似POD加速基于Krylov子空间的迭代线性求解器的收敛

NSPOD: acceleratingthe convergence ofKrylov-based iterative linearsolvers via approximated PODs

作者: Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak, George Em Karniadakis

类别: math.NA, cs.LG

发布日期: 2026-05-08

arXiv ID: 2605.07828v1

链接: http://arxiv.org/abs/2605.07828v1

摘要: The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural operators for specific geometries, but they tend to not perform well on geometries not previously seen during training. We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers, which, despite learning effectively across arbitrary unstructured meshes without requiring retraining, led to only modest reductions in iterations compared to state-of-the-art preconditioners. In this study we introduce Neural Subspace Proper Orthogonal Decomposition (NSPOD), a multigrid-like deep operator network-based preconditioner which can dramatically reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers, even when compared to state-of-the-art methods such as algebraic multigrid preconditioners. We demonstrate its efficiency via numerical experiments on a linearized version of solid mechanics PDEs applied to unstructured domains obtained from complex CAD geometries. We expect that the findings in this study lead to more efficient hybrid preconditioners that can match, or possibly even surpass, the convergence properties of the current gold standard preconditioning methods for solid mechanics PDEs.

4. 神经算子作为高效的函数插值器

Neural Operators as Efficient Function Interpolators

作者: Vasilis Niarchos, Angelos Sirbu, Sokratis Trifinopoulos

类别: cs.LG, cs.AI, math.NA, nucl-th

发布日期: 2026-05-08

arXiv ID: 2605.07792v1

链接: http://arxiv.org/abs/2605.07792v1

摘要: Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator acting by composition on functions of the base-space. Through a range of benchmarks on analytic functions of increasing complexity and dimensionality, we demonstrate that NOs can match or outperform standard multilayer perceptrons and Kolmogorov–Arnold Networks in accuracy while requiring significantly fewer parameters and training time. As a real-world application, we apply a two-dimensional Tensorized Fourier Neural Operator (TFNO) to the nuclear chart, learning a correction to state-of-the-art nuclear mass models as a partially observed residual field. A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches while retaining high parameter efficiency and short training times. More broadly, these results introduce NOs as a scalable framework for finite-dimensional function interpolation, from analytic benchmarks to structured scientific data.

5. 保向数的表示

Direction-Preserving Number Representations

作者: Bardia Zadeh, George A. Constantinides

类别: cs.LG, math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07662v1

链接: http://arxiv.org/abs/2605.07662v1

摘要: Low-precision number formats are widely used in modern machine learning systems due to their efficiency. Accurate direction representation is key to the accuracy of vector operations. This work precisely explores the extent to which the direction of a vector can be represented by selecting its scalar elements from a common finite alphabet of a given size. This is standard practice in machine learning, where low-precision significands may be narrow-width floating-point or integer values. A geometric framework is introduced for analyzing the directional coverage of such product-structured codes. This work analytically quantifies the suboptimality gap between such product-structured codes and spherical codes for the vector as a whole, in both low and asymptotically high dimensions. Furthermore, within the product code class, it is proven that the standard formats of two’s complement, fixed-point, and floating-point are suboptimal, again with quantified gap, pointing to the potential to develop new scalar number formats. Such scalar alphabets are numerically optimized across multiple block dimensions for directional coverage, including the dimension used in NVIDIA’s NVFP4 format. Experimental results are presented comparing the performance of standard formats and the optimized alphabet. We find that for four bits, NVIDIA’s choice of E2M1 closely approximates the optimized alphabet, providing a geometric explanation for its strong performance in low-precision machine learning workloads and an analytical understanding of the link between that superiority and block size. We provide open-source formal proofs in Lean for the theorems in this work, along with the experimental code and the optimized alphabets obtained.

6. 凸通量标量守恒律数值解的柯尔莫哥洛夫 $\varepsilon$-熵

Kolmogorov $\varepsilon$-entropy of numerical solutions for scalar conservation laws with convex flux

作者: Fabio Ancona, Alessio Basti, Fabio Camilli

类别: math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07427v1

链接: http://arxiv.org/abs/2605.07427v1

摘要: Building on the information-theoretic perspective of P.~D.Lax [\textit{Proc.\ Sympos., Math.\ Res.\ Center, Univ.\ Wisconsin}, 1978], we establish a two-sided quantitative compactness estimate for numerical solutions of scalar conservation laws with a uniformly convex flux, expressed in terms of Kolmogorov $\varepsilon$-entropy. We prove that, under specific grid constraints, conservative, monotone finite-difference schemes satisfying a discrete one-sided Lipschitz condition (OSLC) preserve the $1/\varepsilon$ Kolmogorov entropy scaling of the corresponding exact entropy solution set, matching the bounds obtained by DeLellis and Golse [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{58} (2005)] and by Ancona, Glass, and Nguyen [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{65} (2012)]. Specifically, the upper bound follows from the discrete OSLC, while the lower bound relies on a uniform approximation argument on a bounded-variation precursor class. Our results show that prototypical first-order methods are high-resolution in Lax’s sense. Finally, we abstract the lower bound mechanism into a general transfer principle, discuss implications for information recovery via post-processing, and indicate directions for future work.

7. 图上最优传输问题的牛顿法

Newton’s method for optimal transport problem on graphs

作者: Qujiangxue Chen, Jianbo Cui, Luca Dieci, Haomin Zhou

类别: math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07408v1

链接: http://arxiv.org/abs/2605.07408v1

摘要: In this paper, we study dynamical optimal transport on a connected graph from the perspective of the Benamou-Brenier formulation, where densities are assigned to vertices and velocities to edges. However, directly using Newton’s method on the resulting nonlinear systems encounters two potential difficulties: (i) if the graph contains cycles, edge variables are not unique, and (ii) there is no guarantee that the density variables remain positive. To address these challenges, we introduce a finite-difference-type Newton method that eliminates cycle-induced redundancies through a spanning-tree gauge, resulting in a reduced set of independent variables and a well-posed, sparse linear system. For the lattice graph arising from the continuous optimal transport problem, density positivity can also be guaranteed by using an upwind discretization subject to a CFL-type condition. We further demonstrate the versatility of the proposed scheme by applying it to a range of problems, including optimal transport on lattices and random graphs, inverse optimal transport problems, and social network analysis.

8. QuadNorm：分辨率鲁棒的神经算子归一化方法

QuadNorm: Resolution-Robust Normalization for Neural Operators

作者: Bum Jun Kim, Makoto Kawano, Yusuke Iwasawa, Yutaka Matsuo

类别: cs.LG, cs.CE, math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07375v1

链接: http://arxiv.org/abs/2605.07375v1

摘要: Normalization layers in neural operators usually compute statistics by uniformly averaging discrete grid values, making the normalization itself discretization-dependent and thereby a source of transfer error across different resolutions or meshes. To enable discretization robustness, we introduce a quadrature normalization family that replaces existing uniform averaging in normalization layers with numerical quadrature: QuadNorm and BlendQuadNorm. On endpoint-inclusive uniform grids, the proposed quadrature moments are $O(h^2)$-consistent across discretizations, meaning that their cross-resolution mismatch decays quadratically with grid spacing. A transfer-error bound then predicts how normalization-induced mismatch scales with both the resolution gap and network depth. The experiments show the same gap- and depth-scaling trends predicted by the transfer-error bound. On Darcy, QuadNorm delivers the best cross-resolution performance at every tested target resolution from $64^2$ to $256^2$; on real-data benchmarks, Transolver with QuadNorm achieves nearly resolution-invariant transfer. The largest gains appear on nonperiodic PDEs and nonspectral architectures, where native-resolution improvements also emerge. We also validate BlendQuadNorm, which stays close to LayerNorm behavior and serves as a conservative default for periodic FNO settings. These results identify normalization as a previously overlooked source of resolution dependence in neural operators.

9. 利用预条件神经算子求解卷积型积分方程

Solving Convolution-type Integral Equations using Preconditioned Neural Operators

作者: Raymond Chan, Lingfeng Li

类别: math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07365v1

链接: http://arxiv.org/abs/2605.07365v1

摘要: Convolution-type integral equations arise from various fields, \textit{e.g.}, finite impulse response filters in signal processing and deblurring problems in image processing. When solving these equations, conventional numerical methods, like the multigrid method, can only efficiently solve the low-frequency components in the error, but not the high-frequency components. In this paper, we apply neural operators to address this issue. By adopting a preconditioning approach, we propose a novel training strategy that trains neural operators to solve the high-frequency components efficiently. Then, we combine the neural operators with some classical iterative solvers, like the weighted Jacobi method, to obtain an efficient hybrid iterative algorithm for the integral equations. We analyze the generalization error of our training strategy and the convergence of the hybrid iterative algorithm. We test our algorithms on large-scale and ill-conditioned linear systems discretized from one- and two-dimensional convolution-type integral equations. Our proposed algorithm significantly outperforms the multigrid method and the preconditioned conjugate gradient method in both iteration numbers and computational time.

10. Vlasov-Poisson-BGK 模型的欧拉格式的收敛性

Convergence of an Eulerian scheme for the Vlasov-Poisson-BGK model

作者: Seung Yeon Cho, Sungsu Park, Seok-Bae Yun

类别: math.NA, math.AP

发布日期: 2026-05-08

arXiv ID: 2605.07347v1

链接: http://arxiv.org/abs/2605.07347v1

摘要: The Vlasov-Poisson-BGK (VPBGK) model is a kinetic model for describing the dynamics of collisional plasmas. Although various numerical schemes have been developed for it, a corresponding convergence theory has been absent. This paper fills this gap by presenting the first convergence analysis for a non-splitting, finite-difference Eulerian scheme discretized on the full phase-space grid. A major theoretical obstacle is the mixing of velocity indices induced by the electric field, which hinders the derivation of a uniform lower bound for the discrete solution. To overcome this stability challenge, we propose a modified lower bound estimate suitable for ionized systems that incorporates the step-wise degradation. Under a truncated velocity domain with a Neumann boundary condition, we establish error estimates for the distribution function in a weighted $L^{\infty}$ norm and for the electric field in a $L^{\infty}$ norm, respectively.

11. TREA：面向目标检测与分类的低精度时分复用资源高效边缘加速器

TREA: Low-precision Time-Multiplexed, Resource-Efficient Edge Accelerator for Object Detection and Classification

作者: Vijay Pratap Sharma, Mukul Lokhande, Ratko Pilipovic, Omkar Kokane, Santosh Kumar Vishvakarma

类别: cs.AR, cs.DC, eess.IV, math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07321v1

链接: http://arxiv.org/abs/2605.07321v1

摘要: This work presents TREA, a low-precision time-multiplexed and resource-efficient edge-AI accelerator for object detection and classification, targeting stringent area-power-latency constraints of edge vision platforms. The proposed architecture integrates a dual-precision (4/8-bit) SIMD multiply-accumulate (DQ-MAC) unit based on most-significant-digit-first (MSDF) shift-and-add computation with run-time bit truncation, eliminating conventional multiplier overhead and reducing accumulator bit-width. The DQ-MAC supports 4x FxP4 or 1x FxP8 operations per cycle, achieving up to 4x throughput improvement without hardware duplication. A structured hardware-aware reductive pruning (SHARP) strategy is co-designed with the SIMD datapath, enabling near 50% structured sparsity while maintaining full MAC utilization. This allows a 3x3 convolution kernel to be computed in 1 cycle in FxP4 mode compared to 9 cycles in FxP8, and a 5x5 kernel in 3 cycles versus 25 cycles, yielding up to 9x latency reduction at the kernel level. The accelerator further incorporates a reconfigurable CORDIC-based nonlinear activation function (RQ-NAF) core with a 9-stage pipeline, supporting Sigmoid, Tanh, and ReLU at one output per cycle after pipeline fill, while enabling (N-1) hardware reuse through time-multiplexing. The complete TREA architecture employs a 1D array of 100 SIMD DQ-MAC units with layer-wise hardware reuse, significantly reducing area and control complexity. Experimental results demonstrate substantial improvements in latency, hardware utilization, and energy efficiency compared to conventional fixed-precision and non-reconfigurable accelerators, validating TREA as an effective solution for real-time edge vision workloads.

12. 用于奇异摄动常微分方程的基于 Krylov 奇异值分解的稀疏随机特征神经网络

Sparse Random-Feature Neural Networks with Krylov-Based SVD for Singularly Perturbed ODE

作者: Kevin Kurian Thomas Vaidyan, Siddharth Rout

类别: math.NA, cs.LG, physics.comp-ph

发布日期: 2026-05-08

arXiv ID: 2605.07286v1

链接: http://arxiv.org/abs/2605.07286v1

摘要: Random-feature neural networks (RFNNs), including architectures with fixed hidden layers and analytically determined output weights, offer fast training but often suffer from issues due to dense representations of the hidden layer activation. Their reliance on dense feature mappings and least squares solvers can limit scalability and numerical stability, particularly for high-dimensional or stiff systems. Specifically, the activation matrix is observed to be low-rank and extremely ill-conditioned. In this work, we propose a sparse framework for RFNNs that integrates structured sparsity into the hidden layer activations that increases the rank and employs Sparse Singular Value Decomposition (sSVD) for solving the resulting linear least squares problem scalably and efficiently while catering to the bad condition number. We explore the theory behind Lanczos-Golub-Kahan Bidiagonalization technique for sparse SVD and conduct some experiments to identify some limitations and justify the requirement for orthogonalization step in our application. Then, we demonstrate that the proposed method maintains or improves solution accuracy for solving the benchmark one-dimensional steady convection-diffusion equations case having stronger advection, while achieving substantial gains in training efficiency and robustness compared to standard dense implementations.

13. 变量解耦与有理函数的柯尔莫哥洛夫叠加定理

Variable decoupling and the Kolmogorov Superposition Theorem for rational functions

作者: A. C. Antoulas, I. V. Gosea, C. Poussot-Vassal

类别: math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07246v1

链接: http://arxiv.org/abs/2605.07246v1

摘要: This work shows that for rational multivariate functions, the Kolmogorov Superposition Theorem (KST) involves several single-variable functions, which can be written down by inspection. In other words, no computation is required for decoupling the variables of multivariate rational functions. The key tool for this development is the Loewner Framework for multivariate functions. Applications of this result involve approximating multivariate non-rational functions by low-complexity multivariate rational and polynomial functions.

14. 高维线性量子系统的辛H2模型降阶

Symplectic H2 Model Reduction for High-Dimensional Linear Quantum Systems

作者: Alfo Borzi, Guofeng Zhang

类别: quant-ph, eess.SY, math.NA, math.OC

发布日期: 2026-05-08

arXiv ID: 2605.07152v1

链接: http://arxiv.org/abs/2605.07152v1

摘要: The $\mathcal{H}_2$ model reduction problem for high-dimensional linear quantum systems is studied under the constraint of physical realizability (PR). This constraint requires preservation of the canonical commutation relations and the quantum input-output structure, and therefore prevents the direct use of standard projection methods. A symplectic Petrov-Galerkin framework is presented, in which reduced-order models automatically satisfy the PR identities by construction. Within this framework, a symplectic variant of the iterative rational Krylov algorithm is developed and referred to as Quantum IRKA (Q-IRKA). At each iteration, an enriched tangential rational Krylov pool is generated from shifted linear solves. A symplectic basis is then extracted by a Gram-Schmidt-type procedure, paired with symplectic conjugates, and normalized so that the reduced trial space satisfies the canonical symplectic constraint. The interpolation points are updated from selected mirror images of the poles of the current reduced-order model, while the reduced-order matrices are obtained exclusively by structure-preserving projection. Numerical experiments on low-channel oscillator-chain systems and on a bosonic Kitaev-chain-inspired benchmark show that Q-IRKA is effective for large-scale linear quantum systems. Symplecticity and PR are preserved to machine precision, and accurate reduced-order models are obtained with moderate computational cost. The results also show that reduction quality depends substantially on dissipation geometry, channel placement, heterogeneity, and reduced order. These findings indicate that scalable $\mathcal{H}_2$ model reduction of linear quantum systems can be achieved while strictly preserving the underlying physical structure.

15. TriP: 一种面向鲁棒平移平均的三角拼图方法

TriP: A Triangle Puzzle Approach to Robust Translation Averaging

作者: Zhekai Fan, Wanze Li, Jinxin Wang, Yunpeng Shi

类别: cs.CV, cs.RO, math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07143v1

链接: http://arxiv.org/abs/2605.07143v1

摘要: Translation averaging aims to recover camera locations from pairwise relative translation directions and is a fundamental component of global Structure-from-Motion pipelines. The problem is challenging because direction measurements contain no distance information, making the estimation problem highly ill-conditioned and highly sensitive to corrupted observations. In this paper, we propose TriP, a triangle-based framework for robust translation averaging. TriP first infers local relative edge scales from triangle geometry, and then synchronizes the scales of overlapping triangles in the logarithmic domain to recover globally consistent edge lengths and camera locations. By leveraging higher-order consistency across triangles, the proposed method is robust to adversarial, cycle-consistent, and other structured corruptions. In addition, TriP avoids the collapse issue without requiring any extra anti-collapse constraints, since log-scale synchronization excludes the degenerate zero-scale solution by construction. These structural advantages enable a particularly strong theory for exact location recovery. On the practical side, TriP is fully parallelizable, computationally efficient, and naturally scalable to graphs with millions of cameras. Moreover, it outperforms all previous translation averaging methods by a large margin on both synthetic and real datasets.

16. 非线性系统随机列块Bregman方法

On a stochastic column-block bregman method for nonlinear systems

作者: Wendi Bao, Naiyu Jiang, Lili Xing, Weiguo Li

类别: math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07136v1

链接: http://arxiv.org/abs/2605.07136v1

摘要: Sparse solution problems play an important role in both signal processing and image restoration. In this paper, we propose a stochastic column-block nonlinear Bregman method for efficiently computing sparse solutions to nonlinear systems. Under certain assumptions, we analyze the convergence of the proposed method and derive an upper bound for its convergence rate. Numerical experiments, including an image recovery problem, are presented to illustrate the efficiency of the proposed method.

17. 稳定化的神经 Hamilton–Jacobi–Bellman 求解器: 误差分析及在基于模型的强化学习中的应用

Stabilized neural Hamilton–Jacobi–Bellman solvers: Error analysis and applications in model-based reinforcement learning

作者: Minseok Kim, Yeongjong Kim, Namkyeong Cho, Yeoneung Kim

类别: cs.LG, cs.AI, math.NA, math.OC

发布日期: 2026-05-08

arXiv ID: 2605.07116v1

链接: http://arxiv.org/abs/2605.07116v1

摘要: Physics-informed neural solvers offer a promising route to model-based reinforcement learning in continuous time, where optimal feedback synthesis is governed by Hamilton–Jacobi–Bellman (HJB) equations. Practical implementations often occupy a regime that is neither a classical grid method nor a continuous-PDE PINN: the value function is represented by a neural network, finite-difference HJB policy-evaluation operators are evaluated by network queries at shifted points, and residuals are minimized by random continuous collocation. This regime preserves the stabilized finite-difference policy-evaluation structure while avoiding grid-based value unknowns. We develop an error theory for this hybrid regime. Interpreting finite differences as shift operators acting on neural networks, we prove a population $L^2$ stability estimate for one policy-evaluation step with learned dynamics. The bound separates residual error, initial and exterior-collar mismatch, policy mismatch, and model-identification error, with an explicit gradient amplification factor for learned dynamics, while the underlying linear evaluation stability remains free of hidden inverse-viscosity blow-up. We further give a finite-sample collocation certificate and a conditional multi-step propagation result through greedy policy improvement. Experiments on compact-control LQR upto 64 dimensions, Allen–Cahn control, pendulum, Hopper, and 3D quadrotor benchmarks compare against representative model-based and model-free RL baselines, demonstrating the predicted residual, policy-mismatch, and learned-model error trends.

18. 等距约束的近端伽辽金方法

Proximal Galerkin for the isometry constraint

作者: Brendan Keith, Frédéric Marazzato

类别: math.NA

发布日期: 2026-05-08

arXiv ID: 2605.07054v1

链接: http://arxiv.org/abs/2605.07054v1

摘要: We resolve a longstanding open problem in the computational modeling of nonlinear plates by introducing a numerical method that exactly enforces the isometry constraint, namely, that the first fundamental form of the mid-surface coincides with the identity tensor. Several numerical methods have been proposed to approximate solutions of such manifold-constrained variational problems using gradient flows with tangent space updates. However, this class of methods presents two main challenges. First, a preprocessing step is required to enforce the boundary conditions and generate an initial guess sufficiently close to an isometry. Second, each step of the gradient flow typically increases the isometry defect. We adopt an alternative approach based on the proximal Galerkin framework, originally introduced for variational problems with convex inequality constraints. The resulting method preserves the geometric structure of the feasible set and yields an efficient algorithm in which each iterate is an exact isometry at the barycenter of every mesh cell. In contrast to existing methods, no preprocessing step is required, enabling broader applicability of this important category of mathematical models. Numerical experiments on standard benchmarks demonstrate that the method converges to a prescribed error tolerance in an asymptotically mesh-independent number of iterations and requires substantially fewer iterations than previous methods, even on coarse meshes.

19. 利用保物理神经网络加速常微分方程模拟

Accelerating the Simulation of Ordinary Differential Equations Through Physics-Preserving Neural Networks

作者: Andrew Tagg, Andrew Frandsen, Andrew Ning

类别: math.NA

发布日期: 2026-05-07

arXiv ID: 2605.06980v1

链接: http://arxiv.org/abs/2605.06980v1

摘要: Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in order to resolve the solution accurately, resulting in increased computational cost. In order to accelerate the simulation of these ODEs we present a novel methodology that uses a pseudo-invertible neural network to map system states into a high-dimensional latent-space. The network is then trained so that the dynamics in this learned latent space are slow, and can be simulated with relatively few function calls. Unlike existing neural methods, the latent dynamic equations are not learned from trajectory data, but derived from the original system equations and the chain rule. This allows the method to generalize better than existing approaches because the derived equations are correct by construction. In this work, we derive latent state equations of motion for any general ODE, and describe the loss function used to enforce slow time evolution of the latent states. We then apply this technique to multiple example ODEs and show that these problems can be solved with $3$x to $20$x fewer function calls for the same accuracy when simulating in the learned latent space. This reduction in cost could decrease computational demands for scientific simulations across engineering and physics applications.

20. 适用于三次和四次势的高效辛积分器

Efficient symplectic integrators for cubic and quartic potentials

作者: Alejandro Escorihuela-Tomàs

类别: math.NA

发布日期: 2026-05-07

arXiv ID: 2605.06975v1

链接: http://arxiv.org/abs/2605.06975v1

摘要: We present a set of new, efficient high-order symplectic methods designed for Hamiltonian systems with cubic or quartic potentials. By demonstrating that polynomial potentials require fewer order conditions, we develop schemes that outperform both standard symmetric compositions of second-order methods and existing RKN splitting methods. Numerical results confirm their improved efficiency over state-of-the-art alternatives found in the literature.

21. EULER-ADAS：面向精度可重构近似ADAS加速的高能效与SIMD统一对数-Posit引擎

EULER-ADAS: Energy-Efficient & SIMD-Unified Logarithmic-Posit Engine for Precision-Reconfigurable Approximate ADAS Acceleration

作者: Mukul Lokhande, Ratko Pilipovic, Omkar Kokane, Adam Teman, Santosh Kumar Vishvakarma

类别: cs.AR, cs.AI, cs.CV, eess.IV, math.NA

发布日期: 2026-05-07

arXiv ID: 2605.06875v1

链接: http://arxiv.org/abs/2605.06875v1

摘要: Advanced driver-assistance systems (ADAS) require neural compute engines that deliver low-latency inference under strict power and area constraints. Posit arithmetic is attractive for such accelerators because it provides high numerical fidelity at low precision, but its variable-length regime encoding increases encode/decode cost and exposes the datapath to large regime-field fault effects. This paper presents EULER-ADAS, a SIMD-enabled logarithmic bounded-Posit neural compute engine for energyefficient and reliability-aware ADAS acceleration. The proposed datapath combines bounded-regime Posit representation, stageadaptive logarithmic mantissa multiplication with bit truncation, and a SIMD-shared quire accumulation path supporting Posit- (8,0), Posit-(16,1), and Posit-(32,2) execution. The unified architecture enables 4xPosit-8, 2xPosit-16, or 1xPosit-32 operation without duplicating precision-specific hardware. FPGA implementation shows that the proposed configurations reduce LUT count by up to 41.4%, delay by up to 76.1%, and power by up to 71.9% relative to exact Posit neural compute engines, while achieving up to 10x lower energy-delay product than radix-4 Booth-based Posit multipliers. In 28-nm CMOS, the bounded variants occupy 0.013-0.016 mm2 , consume 19.8-22.1 mW, and operate at up to 1.84 GHz. Application-level evaluation across image-classification, ADAS, and edge-inference workloads shows that the evaluated Posit-16 and Posit-32 configurations remain within about 1.5 percentage points of FP32 accuracy. A TinyYOLOv3 prototype on Pynq-Z2 achieves 78 ms latency at 0.29 W and 22.6 mJ/frame, demonstrating the suitability of EULERADAS for low-power real-time ADAS inference.

22. 一个算子应对多种密度：基于神经算子的条件化摊销近似

One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators

作者: Panos Tsimpos, Edoardo Calvello, Ayoub Belhadji, Nicholas H. Nelsen

类别: stat.ML, cs.LG, math.NA

发布日期: 2026-05-07

arXiv ID: 2605.06873v1

链接: http://arxiv.org/abs/2605.06873v1

摘要: Probabilistic conditioning is concerned with the identification of a distribution of a random variable $X$ given a random variable $Y$. It is a cornerstone of scientific and engineering applications where modeling uncertainty is key. This problem has traditionally been addressed in machine learning by directly learning the conditional distribution of a fixed joint distribution. This paper introduces a novel perspective: we propose to solve the conditioning problem by identifying a single operator that maps any joint density to its conditional, thus amortizing over joint-conditional pairs. We establish that the conditioning operator can be approximated to arbitrary accuracy by neural operators. Our proof relies on new results establishing continuity of the conditioning operator over suitable classes of densities. Finally, we learn the conditioning map for a class of Gaussian mixtures using neural operators, illustrating the promise of our framework. This work provides the theoretical underpinnings for general-purpose, amortized methods for probabilistic conditioning, such as foundation models for Bayesian inference.

23. Christoffel-DPS：任意分布下扩散后验采样中的最优传感器布置

Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributions

作者: James Rowbottom, Nick Huang, Carola-Bibiane Schönlieb, Ben Adcock

类别: cs.LG, math.NA

发布日期: 2026-05-07

arXiv ID: 2605.06861v1

链接: http://arxiv.org/abs/2605.06861v1

摘要: State estimation is a critical task in scientific, engineering and control applications. Since the reliability of reconstructions depends on the number and position of sensors, optimal sensor placement (OSP) is essential in scenarios where measurements are sparse and expensive. Classical OSP approaches rely on Gaussian assumptions and are consequently unable to account for the complex distributions encountered in many real-world systems. Generative-model-based reconstruction using sensor guided diffusion posterior sampling (DPS) has emerged as a promising technique for reconstructing states from highly complex distributions. However, existing sensor-selection methods either require unrealistically many sensors or emulate classical OSP, creating a mismatch between modern recovery models with classical OSP tools motivating the need for fundamentally new ideas towards OSP that match the recent advances made in powerful recovery models. We introduce a distribution-free sensor placement framework based on the Christoffel function: a mathematical formulation of optimal sampling and recovery guarantees for posterior sampling with arbitrary sensors and signal distributions, from which we derive a new OSP strategy with non-asymptotic bounds on the number of sensors needed for recovery. We develop Christoffel-DPS, with offline and online variants, instantiating Christoffel sampling for generative models. Christoffel-DPS outperforms Gaussian OSP baselines and existing generative-model placement methods, validating that distribution-free sensing is both theoretically principled and practically superior. The framework is model-agnostic; we demonstrate its application to a range of unconditional DPS and flow-matching models on structurally non-Gaussian benchmarks, showing the efficacy of Christoffel-DPS in low sensor budget regimes.