Xiaochuan Tian

Qiang Du, Xingjie Helen Li, Jianfeng Lu et al.

In this paper, we extend the idea of "geometric reconstruction" to couple a nonlocal diffusion model directly with the classical local diffusion in one dimensional space. This new coupling framework removes interfacial inconsistency, ensures the flux balance, and satisfies energy conservation as well as the maximum principle, whereas none of existing coupling methods for nonlocal-to-local coupling satisfies all of these properties. We establish the well-posedness and provide the stability analysis of the coupling method. We investigate the difference to the local limiting problem in terms of the nonlocal interaction range. Furthermore, we propose a first order finite difference numerical discretization and perform several numerical tests to confirm the theoretical findings. In particular, we show that the resulting numerical result is free of artifacts near the boundary of the domain where a classical local boundary condition is used, together with a coupled fully nonlocal model in the interior of the domain.

Qiang Du, Xiaochuan Tian

Nonlocal gradient operators are basic elements of nonlocal vector calculus that play important roles in nonlocal modeling and analysis. In this work, we extend earlier analysis on nonlocal gradient operators. In particular, we study a nonlocal Dirichlet integral that is given by a quadratic energy functional based on nonlocal gradients. Our main finding, which differs from claims made in previous studies, is that the coercivity and stability of this nonlocal continuum energy functional may hold for some properly chosen nonlocal interaction kernels but may fail for some other ones. This can be significant for possible applications of nonlocal gradient operators in various nonlocal models. In particular, we discuss some important implications for the peridynamic correspondence material models.

Qihao Ye, Xiaochuan Tian, Dong Wang

The classical Fokker-Planck equation (FPE) is a key tool in physics for describing systems influenced by drag forces and Gaussian noise, with applications spanning multiple fields. We consider the fractional Fokker-Planck equation (FFPE), which models the time evolution of probability densities for systems driven by Lévy processes, relevant in scenarios where Gaussian assumptions fail. The paper presents an efficient and accurate numerical approach for the free-space FFPE with constant coefficients and Dirac-delta initial conditions. This method utilizes the integral representation of the solutions and enables the efficient handling of very high-dimensional problems using fast algorithms. Our work is the first to present a high-precision numerical solver for the free-space FFPE with Dirac-delta initial conditions. In addition to Dirac-delta initial data, we demonstrate the effectiveness of our method for initial conditions given by sums of Gaussians. This opens the door for future research on more complex scenarios, including those with variable coefficients and other types of initial conditions.

Zihan Shao, Konstantin Pieper, Xiaochuan Tian

We propose a novel framework for solving nonlinear PDEs using sparse radial basis function (RBF) networks. Sparsity-promoting regularization is employed to prevent over-parameterization and reduce redundant features. This work is motivated by longstanding challenges in traditional RBF collocation methods, along with the limitations of physics-informed neural networks (PINNs) and Gaussian process (GP) approaches, aiming to blend their respective strengths in a unified framework. The theoretical foundation of our approach lies in the function space of Reproducing Kernel Banach Spaces (RKBS) induced by one-hidden-layer neural networks of possibly infinite width. We prove a representer theorem showing that the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. The algorithmic framework is based on a three-phase algorithm to maintain computational efficiency through adaptive feature selection, second-order optimization, and pruning of inactive neurons. Numerical experiments demonstrate the effectiveness of our method and highlight cases where it offers notable advantages over GP approaches. This work opens new directions for adaptive PDE solvers grounded in rigorous analysis with efficient, learning-inspired implementation.

Jihong Wang, Xiaochuan Tian, Zhongqiang Zhang et al.

Data-driven methods have emerged as powerful tools for modeling the responses of complex nonlinear materials directly from experimental measurements. Among these methods, the data-driven constitutive models present advantages in physical interpretability and generalizability across different boundary conditions/domain settings. However, the well-posedness of these learned models is generally not guaranteed a priori, which makes the models prone to non-physical solutions in downstream simulation tasks. In this study, we introduce monotone peridynamic neural operator (MPNO), a novel data-driven nonlocal constitutive model learning approach based on neural operators. Our approach learns a nonlocal kernel together with a nonlinear constitutive relation, while ensuring solution uniqueness through a monotone gradient network. This architectural constraint on gradient induces convexity of the learnt energy density function, thereby guaranteeing solution uniqueness of MPNO in small deformation regimes. To validate our approach, we evaluate MPNO's performance on both synthetic and real-world datasets. On synthetic datasets with manufactured kernel and constitutive relation, we show that the learnt model converges to the ground-truth as the measurement grid size decreases both theoretically and numerically. Additionally, our MPNO exhibits superior generalization capabilities than the conventional neural networks: it yields smaller displacement solution errors in down-stream tasks with new and unseen loadings. Finally, we showcase the practical utility of our approach through applications in learning a homogenized model from molecular dynamics data, highlighting its expressivity and robustness in real-world scenarios.

Qihao Ye, Xiaochuan Tian, Yuhua Zhu

This paper develops a model-based framework for continuous-time policy evaluation (CTPE) in reinforcement learning, incorporating both Brownian and Lévy noise to model stochastic dynamics influenced by rare and extreme events. Our approach formulates the policy evaluation problem as solving a partial integro-differential equation (PIDE) for the value function with unknown coefficients. A key challenge in this setting is accurately recovering the unknown coefficients in the stochastic dynamics, particularly when driven by Lévy processes with heavy tail effects. To address this, we propose a robust numerical approach that effectively handles both unbiased and censored trajectory datasets. This method combines maximum likelihood estimation with an iterative tail correction mechanism, improving the stability and accuracy of coefficient recovery. Additionally, we establish a theoretical bound for the policy evaluation error based on coefficient recovery error. Through numerical experiments, we demonstrate the effectiveness and robustness of our method in recovering heavy-tailed Lévy dynamics and verify the theoretical error analysis in policy evaluation.