Special Session 47: Meeting Point of Scientific Computing and Machine Learning

Neural Operator for Multidisciplinary Engineering Design

Daniel Zhengyu Huang Peking University Peoples Rep of China Co-Author(s):

Abstract: Deep learning surrogate models have shown significant promise in solving partial differential equations. These efficient models enable many-query computations in science and engineering, with particular focus on engineering design optimization, which is the central topic of this talk. I will begin by introducing the neural operator approach for surrogate modeling, followed by a theoretical analysis of Bayesian nonparametric regression of linear functionals to better understand the sample complexity.

Deep learning solvers for a couple of fluid dynamic equations

Liwei Xu University of Electronic Science and Technology of China Peoples Rep of China Co-Author(s):

Abstract: In this talk, we present two hybrid methods for solving fluid dynamic equations, which combine neural network methods and traditional methods. The first is applying the asymptotic-preserving and positive-preserving techniques to physics-informed neural networks, which has advantages in high dimensionality and multiscale characteristics. The second is learning a class of new discretization schemes with neural networks, which has high accuracy in the smooth stencil and maintains essentially non-oscillation near the discontinuity. Numerical simulations are employed to validate our methods.

pETNNs: Partial Evolutionary Tensor Neural Networks for Solving Time-dependent Partial Differential Equations

Jin Zhao Capital Normal University Peoples Rep of China Co-Author(s):    Tunan Kao, He Zhang, and Lei Zhang

Abstract: In this talk, we will introduce our recent work for solving time-dependent partial differential equations with both of high accuracy and remarkable extrapolation, called partial evolutionary tensor neural networks (pETNNs). Our proposed architecture leverages the inherent accuracy of tensor neural networks, while incorporating evolutionary parameters that enable remarkable extrapolation capabilities. By adopting innovative parameter update strategies, the pETNNs achieve a significant reduction in computational cost while maintaining precision and robustness. Notably, the pETNNs enhance the accuracy of conventional evolutional deep neural networks and empowers computational abilities to address high-dimensional problems. Numerical experiments demonstrate the superior performance of the pETNNs in solving time-dependent complex equations, including the Navier-Stokes equations, high-dimensional heat equation, high-dimensional transport equation and Korteweg-de Vries type equation.

On finite element approximation of the Schroedinger-Poisson model

Weying Zheng Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Tao Cui, Wenhao Lu, and Naiyan Pan

Abstract: In this paper, we study the finite element approximation of the nonlinear Schr\{o}dinger-Poisson model. The electron density is defined by an infinite series over all eigenvalues of the Hamiltonian operator. To establish the error estimate, we present an abstract theory of error estimates for a class of nonlinear problems. The nonlinear problem is first formulated as a fixed-point equation of a compact mapping $\mathcal{A}$. By constructing an approximate mapping $\mathcal{A}_h$, we prove that $\mathcal{A}_h$ has a fixed point $u_h$ which is the solution to the nonlinear approximate problem. The error estimate between $u$ and $u_h$ is established. We apply the abstract theory to the finite element approximation of the Schr\{o}dinger-Poisson model and obtain optimal error estimate between the numerical solution and the exact solution. Numerical experiments are presented to verify the convergence rates of numerical solutions.

Numerical analysis for manifold-preserving and data-driven algorithms of high-index saddle dynamics

Xiangcheng Zheng Shandong University Peoples Rep of China Co-Author(s):    Lei Zhang, Pingwen Zhang, Xiangcheng Zheng

Abstract: High-index saddle dynamics (HiSD) is a powerful instrument in finding multiple saddle points of complex systems. A critical point in designing numerical algorithms is to preserve the manifold properties of HiSD. We perform numerical analysis for manifold-preserving numerical approximation to HiSD, which not only gives error estimates but provides expatiation for manifold-preserving mechanisms of the continuous HiSD. Furthermore, a data-driven HiSD algorithm is presented and analyzed to improve the applicability of the HiSD.