Journal of Computational Physics最新文献_第7页

R-PINN: Recovery-type a-posteriori estimator enhanced adaptive PINN R-PINN：恢复型后验估计器，增强自适应PINN

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-14 DOI: 10.1016/j.jcp.2026.114684

Rongxin Lu , Jiwei Jia , Young Ju Lee , Zheng Lu , Chen-Song Zhang

In recent years, with the advancements in machine learning and neural networks, algorithms using physics-informed neural networks (PINNs) to solve PDEs have gained widespread applications. While these algorithms are well-suited for a wide range of equations, they often exhibit a suboptimal performance when applied to equations with large local gradients, resulting in substantially localized errors. To address this issue, this paper proposes an adaptive PINN algorithm designed to improve accuracy in such cases. The core idea of the algorithm is to adaptively adjust the distribution of collocation points based on the recovery-type a-posteriori error of the current numerical solution, enabling a better approximation of the true solution. This approach is inspired by the adaptive finite element method. By combining the recovery-type a-posteriori estimator, a gradient-recovery estimator commonly used in the adaptive finite element method (FEM), with PINNs, we introduce the recovery-type a-posteriori estimator enhanced adaptive PINN (R-PINN) and compare its performance with a typical adaptive sampling PINN, failure-informed PINN (FI-PINN), and a typical adaptive weighting PINN, residual-based attention in PINN (RBA-PINN) as a baseline. Our results demonstrate that R-PINN achieves faster convergence with fewer adaptively distributed points and outperforms the other two PINNs in the cases with regions of large errors.

近年来，随着机器学习和神经网络的发展，利用物理信息神经网络（pinn）求解偏微分方程的算法得到了广泛的应用。虽然这些算法非常适合于广泛的方程，但当应用于具有大局部梯度的方程时，它们往往表现出次优的性能，导致大量的局部误差。为了解决这个问题，本文提出了一种自适应PINN算法，旨在提高这种情况下的准确性。该算法的核心思想是基于当前数值解的恢复型后验误差自适应调整配点的分布，使其能够更好地逼近真实解。该方法受到自适应有限元法的启发。通过将自适应有限元法（FEM）中常用的梯度恢复估计（recovery-type a- posterori estimator）与PINN相结合，引入了增强自适应PINN (R-PINN)，并将其性能与典型的自适应采样PINN （FI-PINN）和典型的自适应加权PINN （RBA-PINN）作为基准进行了比较。研究结果表明，R-PINN在自适应分布点较少的情况下收敛速度更快，在误差区域较大的情况下优于其他两种pinn。

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引用次数: 0

Enabling probabilistic learning on manifolds through double diffusion maps 通过双扩散映射实现流形的概率学习

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-09 DOI: 10.1016/j.jcp.2026.114663

Dimitris G. Giovanis , Nikolaos Evangelou , Ioannis G. Kevrekidis , Roger G. Ghanem

We present a generative learning framework for probabilistic sampling that extends Probabilistic Learning on Manifolds (PLoM), which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a (representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to characterize the manifold of the data, and (c) a reduced-order Itô Stochastic Differential Equation (ISDE) to sample from the learned distribution. However, its sampling dynamics are posed in the ambient space and the retained number of reduced coordinates is chosen by projection-reconstruction error. In practice, this often (i) requires more coordinates than the data’s intrinsic dimension to achieve stable sampling and (ii) lacks a smooth, basis-independent lifting back to the data domain; moreover, standard Diffusion Maps emphasize harmonic eigenfunctions and can miss non-harmonic latent structure. We address these limitations by decoupling geometry learning from sampling: a first Diffusion Maps pass identifies non-harmonic coordinates on which we formulate a full-order ISDE directly in the latent space, while Double Diffusion Maps captures multiscale geometric features and Geometric Harmonics (GH) learns a smooth lifting map to the ambient variables that is independent of the particular diffusion basis. This hybrid design preserves the system’s dynamical richness with a compact geometric representation and enables principled out-of-sample inference. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

我们提出了一个概率抽样的生成式学习框架，扩展了流形上的概率学习（PLoM），该框架旨在生成有限维欧几里得空间中随机向量的统计一致实现，由一组（代表性）观察结果提供信息。在其原始形式中，PLoM通过结合三个主要组成部分构建了一个降阶概率模型：(a)核密度估计来近似潜在的概率度量，(b)扩散映射来表征数据的流形，以及(c)一个降阶Itô随机微分方程（ISDE）来从学习分布中采样。然而，它的采样动态是在环境空间中进行的，并通过投影重建误差来选择保留的约简坐标数。在实践中，这通常(i)需要比数据的固有维度更多的坐标来实现稳定的采样，（ii）缺乏平滑的、与基无关的提升回数据域；此外，标准扩散图强调谐波特征函数而忽略非谐波潜在结构。我们通过从采样中解耦几何学习来解决这些限制：第一次扩散地图通过识别非调和坐标，我们直接在潜在空间中制定全阶ISDE，而双扩散地图捕获多尺度几何特征，几何谐波（GH）学习到独立于特定扩散基础的环境变量的平滑提升图。这种混合设计通过紧凑的几何表示保留了系统的动态丰富性，并使原则的样本外推理成为可能。通过两个数值研究证明了该方法的有效性和鲁棒性：一个是基于二维Hermite多项式函数生成的数据，另一个是基于高保真的反应流爆震波模拟。

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引用次数: 0

Toward accurate and efficient multi-moment finite volume method for large eddy simulations of compressible flows on unstructured grids 非结构网格上可压缩流大涡模拟的精确、高效多矩有限体积法

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-09 DOI: 10.1016/j.jcp.2026.114668

Ying Yang , Feng Xiao , Bin Xie

A novel multi-moment finite volume method is proposed for the solutions of linear and nonlinear hyperbolic equations on unstructured grids and further investigated for implicit large eddy simulation of compressible turbulence. Different from the previous volume integrated average and point value based multi-moment (VPM) method, the present scheme first reconstructs the solution values and first-order derivatives at boundary surfaces and then constructs a quadratic polynomial over each cell using Gauss divergence theorem. It eliminates the need to directly compute second derivatives of solution variables, which is more efficient than conventional finite volume discretizations at third-order accuracy. The so-called VPM-FR (VPM with face-based reconstruction) formulates novel spatial reconstruction within a compact stencil consisting of only face neighbouring cells, which substantially improves the numerical accuracy, efficiency, robustness as well as algorithmic simplicity. Fourier analysis is also conducted to verify the numerical properties of VPM-FR which are compared against the previous version of VPM scheme. Besides, a new limiting projection approach is devised to use a high-order limiter function which effectively suppresses the numerical oscillation in the vicinity of discontinuities. Numerical results of various benchmark tests are presented for the advection, Euler and Navier-Stokes equations which validate the excellent performance of VPM-FR scheme that well resolves broadband turbulence and the sharp shock profiles with a reduction in computation cost.

提出了求解非结构网格上线性和非线性双曲型方程的一种新的多矩有限体积法，并进一步研究了可压缩湍流的隐式大涡模拟。与以往的体积积分平均和基于点值的多矩（VPM）方法不同，该方法首先在边界面上重建解值和一阶导数，然后利用高斯散度定理在每个单元上构造二次多项式。它消除了直接计算解变量二阶导数的需要，这比传统的三阶精度有限体积离散更有效。所谓的vvm - fr （VPM with face-based reconstruction）在仅由人脸相邻单元组成的紧凑模板内制定了新颖的空间重建，这大大提高了数值精度，效率，鲁棒性以及算法的简单性。通过傅里叶分析验证了VPM- fr格式的数值特性，并与之前版本的VPM格式进行了比较。此外，设计了一种新的极限投影方法，利用高阶极限函数有效地抑制了不连续点附近的数值振荡。对平流方程、Euler方程和Navier-Stokes方程进行了各种基准测试，验证了vvm - fr格式在解决宽带湍流和剧烈激波剖面问题上的优异性能，并降低了计算成本。

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引用次数: 0

Differentiable neural network representation of multi-well, locally-convex potentials 多井，局部凸电位的可微神经网络表示

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-14 DOI: 10.1016/j.jcp.2026.114688

Reese E. Jones , Adrian Buganza Tepole , Jan N. Fuhg

Multi-well potentials are ubiquitous in science, modeling phenomena such as phase transitions, dynamic instabilities, and multimodal behavior across physics, chemistry, and biology. In contrast to non-smooth minimum-of-mixture representations, we propose a differentiable and convex formulation based on a log-sum-exponential (LSE) mixture of input convex neural network (ICNN) modes. This log-sum-exponential input convex neural network (LSE-ICNN) provides a smooth surrogate that retains convexity within basins and allows for gradient-based learning and inference.

A key feature of the LSE-ICNN is its ability to automatically discover both the number of modes and the scale of transitions through sparse regression, enabling adaptive and parsimonious modeling. We demonstrate the versatility of the LSE-ICNN across diverse domains, including mechanochemical phase transformations, microstructural elastic instabilities, conservative biological gene circuits, and variational inference for multimodal probability distributions. These examples highlight the effectiveness of the LSE-ICNN in capturing complex multimodal landscapes while preserving differentiability, making it broadly applicable in data-driven modeling, optimization, and physical simulation.

多井势在科学中无处不在，可以模拟物理、化学和生物学中的相变、动态不稳定性和多模态行为等现象。与非光滑混合最小表示相反，我们提出了一种基于输入凸神经网络（ICNN）模式的对数和指数（LSE）混合的可微凸公式。这种对数和指数输入凸神经网络（LSE-ICNN）提供了一个平滑的代理，保留了盆地内的凸性，并允许基于梯度的学习和推理。LSE-ICNN的一个关键特征是它能够通过稀疏回归自动发现模式的数量和转换的规模，从而实现自适应和简约的建模。我们展示了LSE-ICNN在不同领域的多功能性，包括机械化学相变、微观结构弹性不稳定性、保守的生物基因回路和多模态概率分布的变分推理。这些例子突出了LSE-ICNN在捕获复杂多模态景观的同时保持可微分性的有效性，使其广泛适用于数据驱动的建模、优化和物理模拟。

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引用次数: 0

Investigation of new analytical and numerical solutions of the extended (2+1) dimensional Boussinesq equation using fractional derivative approaches 用分数阶导数方法研究扩展（2+1）维Boussinesq方程新的解析解和数值解

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-11 DOI: 10.1016/j.jcp.2026.114654

İlknur Kızıl , Ulviye Demirbilek , Ercan Çelik

This study investigates the new fractional (2+1)-dimensional, extended Boussinesq (eBO) equation, which models the behavior of shallow water waves in channels with constant depth and flow velocity. This equation holds considerable relevance in fields such as ocean engineering, coastal hydrodynamics, plasma physics, and nonlinear wave theory. By applying the

(m + 1 / G^{'})

–expansion method, the sub-equation method, and dummyTXdummy– the modified extended tanh function method within the framework of conformable derivatives, a wide array of analytical traveling wave solutions is obtained. These include dark, bright, periodic, singular, exponential, and generalized hyperbolic-type solitons with kink-like features. The fractional transformation technique transforms the original fractional partial differential equations into ordinary differential equations, thereby simplifying the solution process. Moreover, the residual power series method (RPSM) is employed to approximate the solution of this equation, and modulation instability (MI) analysis is conducted to evaluate the stability of the obtained analytical solutions. The study includes comparison tables and various graphical representations to validate the solutions. The numerical findings demonstrate that all methods effectively provide exact and approximate solutions to nonlinear fractional differential equations. The temporal progression and spatial features of the solutions are visualized through 2D, 3D, and contour plots with a comparison of different fractional values. Computational validations are conducted using software, demonstrating the efficiency and general applicability of the proposed approach.

本文研究了新的分数（2+1）维扩展Boussinesq （eBO）方程，该方程模拟了浅水波在恒定深度和恒定流速的通道中的行为。该方程在海洋工程、海岸流体力学、等离子体物理和非线性波浪理论等领域具有相当大的相关性。采用（m+1/G’）展开法、子方程法和dummyTXdummy -改进的扩展tanh函数法，在适形导数的框架下，得到了广泛的行波解析解。这些孤子包括暗孤子、亮孤子、周期孤子、奇异孤子、指数孤子和具有扭结样特征的广义双曲型孤子。分数阶变换技术将原来的分数阶偏微分方程转化为常微分方程，从而简化了求解过程。利用残差幂级数法（RPSM）逼近方程的解，并进行了调制不稳定性分析（MI）来评价所得解析解的稳定性。该研究包括比较表和各种图形表示来验证解决方案。数值结果表明，所有方法都能有效地提供非线性分数阶微分方程的精确近似解。通过2D、3D和等高线图来可视化解决方案的时间进展和空间特征，并对不同分数值进行比较。利用软件进行了计算验证，证明了所提出方法的效率和一般适用性。

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引用次数: 0

An implicit shock tracking method for simulation of shock-dominated flows over complex domains using mesh-based parametrizations 基于网格参数化的复杂区域激波主导流动模拟的隐式激波跟踪方法

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-09 DOI: 10.1016/j.jcp.2025.114647

Alexander M. Pérez Reyes, Matthew J. Zahr

A mesh-based parametrization is a parametrization of a geometric object that is defined solely from a mesh of the object, e.g., without an analytical expression or computer-aided design (CAD) representation of the object. In this work, we propose a mesh-based parametrization of an arbitrary d′-dimensional object embedded in a d-dimensional space using tools from high-order finite elements. Using mesh-based parametrizations, we construct a boundary-preserving parametrization of the nodal coordinates of a computational mesh that ensures all nodes remain on all their original boundaries. These boundary-preseving parametrizations allow the nodes of the mesh to move only in ways that will not change the computational domain. They also ensure nodes will not move between boundaries, which would cause issues assigning boundary conditions for partial differential equation simulations and lead to inaccurate geometry representations for non-smooth boundary transitions. Finally, we integrate boundary-preserving, mesh-based parametrizations into high-order implicit shock tracking, an optimization-based discontinuous Galerkin method that moves nodes to align mesh faces with non-smooth flow features to represent them perfectly with inter-element jumps, leaving the intra-element polynomial basis to represent smooth regions of the flow with high-order accuracy. Mesh-based parametrizations enable implicit shock tracking simulations of shock-dominated flows over geometries without simple analytical parametrizations. Several demonstrations of mesh-based parametrizations are provided to: (1) give concrete examples of the formulation, (2) show that accurate parametrizations can be obtained despite the surrogate surfaces only being C⁰, (3) show they integrate seemlessly with implicit shock tracking and can be used to parametrize surfaces without explicit expressions, and (4) effectively parametrize complex geometries and prevent nodes from moving off their original boundaries.

基于网格的参数化是一种几何对象的参数化，它仅从对象的网格中定义，例如，不使用对象的解析表达式或计算机辅助设计（CAD）表示。在这项工作中，我们提出了一种基于网格的参数化方法，该方法使用高阶有限元工具对嵌入在d维空间中的任意d维对象进行参数化。使用基于网格的参数化，我们构建了计算网格的节点坐标的边界保持参数化，以确保所有节点保持在所有原始边界上。这些保持边界的参数化允许网格节点仅以不会改变计算域的方式移动。它们还确保节点不会在边界之间移动，这将导致为偏微分方程模拟分配边界条件的问题，并导致非光滑边界转换的不准确几何表示。最后，我们将基于边界保持的网格参数化集成到高阶隐式激波跟踪中，这是一种基于优化的不连续伽辽金方法，该方法移动节点使网格面与非光滑流动特征对齐，以单元间跳跃完美地表示它们，留下单元内多项式基以高阶精度表示流动的光滑区域。基于网格的参数化可以在没有简单分析参数化的情况下对几何形状的激波主导流动进行隐式激波跟踪模拟。提供了几个基于网格的参数化的演示：(1)给出了公式的具体示例；(2)表明，尽管代理曲面只有C0，但可以获得准确的参数化；(3)表明它们与隐式冲击跟踪无缝集成，可以用于参数化曲面，而不需要显式表达式；(4)有效地参数化复杂几何形状并防止节点偏离其原始边界。

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引用次数: 0

Polynomial range estimation as a troubled-cell indicator for high-order methods 多项式距离估计作为高阶方法的故障单元指示器

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-14 DOI: 10.1016/j.jcp.2026.114687

Madeline M. Peck, Jiajia Waters

Two troubled-cell indicators based on polynomial range estimation methods are used to flag cells that may violate positivity constraints. One method uses interval extension, and the second uses the range enclosure property of the Bernstein polynomial basis. Both methods reduce compute time for the positivity preserver by limiting its application to a subset of cells. The Bernstein polynomial method remains effective as the problem dimensionality increases. Interval extension applied to the internal energy equation permits the use of the troubled-cell indicators for rational functions, though performance suffers compared to directly applying the indicators to polynomial functions.

采用基于多项式距离估计方法的两种故障单元指示器来标记可能违反正性约束的单元。一种方法使用区间扩展，另一种方法使用Bernstein多项式基的范围封闭性质。这两种方法都通过限制其应用于细胞子集来减少计算时间。随着问题维数的增加，Bernstein多项式方法仍然有效。应用于内能方程的区间扩展允许对有理函数使用故障单元指示器，尽管与直接将指示器应用于多项式函数相比，性能会受到影响。

引用次数: 0

Three-dimensional soft discrete element method for large-scale simulations of soft spheres 三维软离散元法在软球体大尺度模拟中的应用

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-12 DOI: 10.1016/j.jcp.2026.114681

Zonglin Li , Ju Chen , Qiang Tian , Haiyan Hu

Soft particles are ubiquitous in both nature and industries, yet existing three-dimensional simulation methods remain inefficient. This paper presents a three-dimensional soft discrete element method (3D SDEM) for the efficient large-scale dynamic simulations of soft spherical particles. In the method, each soft sphere is modeled as a truncated ellipsoid with a homogeneous strain field, which requires 12 degrees of freedom only. The dynamic equations of soft spheres are derived by using the Lagrange-d’Alembert principle, and the contact detection between soft spheres is handled via the Common Normal Method. The contact force model and the strain energy function are formulated and validated through various compression scenarios of a single soft sphere. The accuracy and efficiency of the 3D SDEM are verified through two-sphere collision simulations and three-dimensional soft sphere compaction simulations. Notably, the compression of 10,000 soft spheres from a jammed state to 95 % volume fraction is simulated within five hours on a laptop computer with a single GPU only. Finally, the method is used to simulate the shear flow of soft particle glasses comprising 1000 soft spheres and successfully capture individual soft sphere deformations not reported before. These results demonstrate that the 3D SDEM enables the efficient modeling of large-scale soft sphere systems, paving the way for advanced studies in both physics and engineering applications.

软粒子在自然界和工业中无处不在，但现有的三维模拟方法仍然效率低下。本文提出了一种三维软离散元法（3D SDEM），用于软球形颗粒的大规模动态模拟。该方法将每个软球建模为具有均匀应变场的截断椭球体，只需要12个自由度。利用拉格朗日-达朗贝尔原理推导了软球的动力学方程，利用公法向法处理了软球之间的接触检测。建立了接触力模型和应变能函数，并通过对单个软球的不同压缩情况进行了验证。通过双球碰撞仿真和三维软球压实仿真，验证了三维SDEM的精度和效率。值得注意的是，在一台只有一个GPU的笔记本电脑上，可以在5小时内模拟1万个软球体从堵塞状态压缩到95%的体积分数。最后，利用该方法模拟了由1000个软球组成的软颗粒玻璃的剪切流动，成功捕获了以前未报道的单个软球变形。这些结果表明，三维SDEM能够有效地建模大型软球系统，为物理和工程应用的深入研究铺平了道路。

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引用次数: 0

Linear high order finite difference methods with essentially non-oscillatory limiters for hyperbolic conservation laws 双曲型守恒律的线性高阶有限差分法本质上是非振荡限制

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-16 DOI: 10.1016/j.jcp.2026.114686

Zhengfu Xu

For high order finite difference and finite volume methods solving hyperbolic conservation laws, the major challenge is to achieve nonlinear stability in the presence of discontinuous solutions. Total variation diminishing or total variation bounded flux limiters are normally set up to achieve the nonlinear stability. High order essentially non-oscillatory methods (ENO or weighted ENO) were designed to avoid constructing high order polynomials across discontinuities to ensure nonlinear stability. However, adaptively reconstructing high order polynomials and doing so in the characteristic space often contributes significantly to the overall computational cost.

Alternatives were proposed as hybrid approaches: simply put, applying limiters in the discontinuous regions of the solution while using linear high order methods in the smooth regions. The key to the success of the hybrid approach lies in the differentiation between smooth and nonsmooth regions, which is highly nontrivial given discrete data sets. In this paper, an irregularity detecting mechanism is provided along the discrete profile of the solution to determine when the nonlinear ENO or WENO methods are needed. The irregularity detector does not depend on manually adjusted parameters when problems change. Such an irregularity detector is easy to implement in the dimensional splitting setting of the finite difference methods. The numerical evidences demonstrate the performance of the newly defined irregularity detector. When applied to high order finite difference methods, the numerical results are accurate and non-oscillatory with improved efficiency.

对于求解双曲型守恒律的高阶有限差分和有限体积方法，主要的挑战是在不连续解存在的情况下实现非线性稳定性。为了达到非线性稳定性，通常设置总变差递减或总变差有界磁通限制器。设计了高阶本质非振荡方法（ENO或加权ENO），避免在不连续点上构造高阶多项式，以保证非线性稳定性。然而，自适应重构高阶多项式并在特征空间中进行自适应重构往往会增加总体计算成本。替代方案被提出为混合方法：简单地说，在解的不连续区域应用限制，而在光滑区域使用线性高阶方法。混合方法成功的关键在于平滑区域和非光滑区域之间的区分，这在给定离散数据集时是非平凡的。在本文中，沿着解的离散轮廓提供了一个不规则检测机制，以确定何时需要非线性ENO或WENO方法。当问题发生变化时，不规则检测器不依赖于手动调整参数。在有限差分法的分维设置下，这种不规则检测方法易于实现。数值证明了新定义的不规则检测器的性能。将其应用于高阶有限差分法时，计算结果准确且无振荡，提高了计算效率。

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引用次数: 0

BiLO: Bilevel Local Operator Learning for PDE Inverse Problems PDE反问题的双层局部算子学习

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2026-04-15 Epub Date: 2026-01-11 DOI: 10.1016/j.jcp.2026.114679

Ray Zirui Zhang , Christopher E. Miles , Xiaohui Xie , John S. Lowengrub

We propose a new neural network based method for solving inverse problems for partial differential equations (PDEs) by formulating the PDE inverse problem as a bilevel optimization problem. At the upper level, we minimize the data loss with respect to the PDE parameters. At the lower level, we train a neural network to locally approximate the PDE solution operator in the neighborhood of a given set of PDE parameters, which enables an accurate approximation of the descent direction for the upper level optimization problem. The lower level loss function includes the least-square penalty of both the residual and its derivative with respect to the PDE parameters. We apply gradient descent simultaneously on both the upper and lower level optimization problems, leading to an effective and fast algorithm. The method, which we refer to as BiLO (Bilevel Local Operator learning), is also able to efficiently infer unknown functions in the PDEs through the introduction of an auxiliary variable. We provide a theoretical analysis that justifies our approach. Through extensive experiments over multiple PDE systems, we demonstrate that our method enforces strong PDE constraints, is robust to sparse and noisy data, and eliminates the need to balance the residual and the data loss, which is inherent to the soft PDE constraints in many existing methods.

将偏微分方程反问题表述为双层优化问题，提出了一种基于神经网络的求解偏微分方程反问题的新方法。在上层，我们将相对于PDE参数的数据丢失最小化。在较低的层次上，我们训练了一个神经网络来局部逼近PDE解算子在给定PDE参数集的邻域，这使得能够精确地逼近上层优化问题的下降方向。较低水平的损失函数包括残差及其导数相对于PDE参数的最小二乘惩罚。我们将梯度下降同时应用于上层和下层的优化问题，从而得到了一个高效、快速的算法。该方法，我们称之为BiLO（双层局部算子学习），也能够通过引入辅助变量有效地推断pde中的未知函数。我们提供了一个理论分析来证明我们的方法是正确的。通过对多个PDE系统的大量实验，我们证明了我们的方法执行强PDE约束，对稀疏和噪声数据具有鲁棒性，并且消除了许多现有方法中软PDE约束所固有的残差和数据丢失平衡的需要。

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引用次数: 0