Special Session 63: Interdisciplinary Applications of Traditional Numerical Methods, Deep Learning Methods, and Statistical Approaches

Well-balanced positivity-preserving high-order discontinuous Galerkin methods for Euler equations with gravitation

Jie Du East China Normal University Peoples Rep of China Co-Author(s):    Yu Wang, Yang Yang and Fangyao Zhu

Abstract: We develop high order discontinuous Galerkin (DG) methods with Lax-Friedrich fluxes for Euler equations under gravitational fields. Such problems may yield steady-state solutions and the density and pressure are positive. There were plenty of previous works developing either well-balanced (WB) schemes to preserve the steady states or positivity-preserving (PP) schemes to get positive density and pressure. However, it is rather difficult to construct WB PP schemes with Lax-Friedrich fluxes, due to the penalty term in the flux. In fact, for general PP DG methods, the penalty coefficient must be sufficiently large, while the WB scheme requires that to be zero. This contradiction can hardly be fixed following the original design of the PP technique. In this talk, we reformulate the source term such that it balances with the flux term when the steady state has reached. To obtain positive numerical density and pressure, we choose a special penalty coefficient in the Lax-Friedrich flux, which is zero at the steady state. The technique works for general steady-state solutions. Numerical experiments are given to show the performance of the proposed methods.

Fast solvers for numerical simulations with applications in microfluidics and antenna design

Shihua Gong The Chinese University of Hong Kong, Shenzhen, SICIAM and SLAI Peoples Rep of China Co-Author(s):

Abstract: Many modern technologies - such as wireless communication devices and microfluidic chips - rely on large-scale numerical simulations. In these simulations, solving the underlying algebraic systems often takes most of the computing time. In this talk, I will introduce several ideas that help accelerate two types of simulations arising from antenna electromagnetics and microfluidic flows.

A GPU-Accelerated Matrix-Free FAS Multigrid Solver with Memory-Efficient Implementations

Zhenlin Guo Beijing Computational Science Research Center Peoples Rep of China Co-Author(s):

Abstract: We develop a matrix-free Full Approximation Storage (FAS) multigrid solver based on staggered finite differences and implemented on GPU in MATLAB. To enhance performance, intermediate variables are reused, and an X-shape Multi-Color Gauss-Seidel (X-MCGS) smoother is introduced. This scheme eliminates the conditional branching required to distinguish red and black nodes in the standard two-color Red-Black Gauss-Seidel (RBGS) method by partitioning the grid into four submatrices according to row-column parity. In addition, restriction and prolongation operators are implemented with GPU acceleration.

Well-balanced and entropy-stable nodal DG method for the Euler equations with gravity

Yan Jiang University of Science and Technology of China Peoples Rep of China Co-Author(s):

Abstract: We propose an entropy stable and well-balanced discontinuous Galerkin (DG) scheme for the Euler equations with gravity To achieve these properties, we utilize the nodal DG framework and carefully design the source term discretization using entropy conservative fluxes. In addition, the proposed formulation is carefully designed to ensure compatibility with a positivity-preserving limiter. We provide a rigorous theoretical analysis to establish the accuracy and structure- preserving properties of the proposed scheme. Extensive numerical experiments confirm the robustness and efficiency of the scheme.

Energy Dissipation Preservation of Implicit-Explicit Linear Multistep Methods for Gradient Flows

Chaoyu Quan The Chinese University of Hong Kong (Shenzhen) Peoples Rep of China Co-Author(s):

Abstract: This paper proposes a theoretical framework for establishing the energy dissipation of general implicit-explicit linear multistep methods (IMEX-LMMs) for gradient flows, by constructing a dissipative modified energy consisting of the original energy and a non-negative quadratic modification. We first test IMEX-LMMs with the first-order time difference of numerical solutions. Then, it is shown that the associated non-negative quadratic modification can be constructed if and only if two generating polynomials (corresponding to the LMM) are positive on $[-1,1]$. Based on this, the modified energy is proved to decay over time under a mild time-step restriction depending on the lower bounds of the associated generating polynomials. As a consequence, the energy dissipation of the well-known backward differentiation formula methods up to fifth order can be obtained straightforwardly. Furthermore, we construct for the first time (to the best of our knowledge) a sixth-order energy-dissipative IMEX-LMM and also prove the sixth-order barrier of energy-dissipative IMEX-LMMs. Some numerical experiments are conducted to verify our theoretical results.

Randomized Neural Networks for PDEs

Fei Wang Xi`an Jiaotong University Peoples Rep of China Co-Author(s):

Abstract: Traditional numerical methods are mathematically rigorous, highly accurate, and physically conservative, forming the reliable foundation of modern scientific computing. Despite their success, they still face intrinsic challenges such as the need for complex mesh generation, limited flexibility in representing global structures, case-by-case reassembly for new configurations, the curse of dimensionality, and difficulties in integrating data or uncertainty. Neural network-based numerical methods have recently emerged as an alternative paradigm, aiming to overcome these limitations through their strong representation capability. However, traditional training-based neural approaches often suffer from the challenges of nonlinear, nonconvex optimization, resulting in limited accuracy and efficiency. To address these issues, we propose a family of randomized neural network (RaNN) methods that integrate the mathematical rigor of classical numerical schemes with the flexibility of neural representations. The framework includes the RaNN-Petrov-Galerkin (RaNN-PG), Local RaNN-Discontinuous Galerkin (LRaNN-DG), LRaNN-HDPG, and LRaNN-Finite Difference methods. Furthermore, an Adaptive-Growth RaNN (AG-RaNN) strategy is introduced to optimize the weighting and selection of random parameters dynamically. We also explore RaNNs for accelerating operator learning, enabling faster training for parameterized PDEs. Numerical experiments demonstrate that RaNN-based methods achieve high accuracy with few degrees of freedom while remaining mesh-free, structure-preserving, and easily extendable to high-dimensional and time-dependent problems. These results highlight RaNNs as a promising direction for unifying classical numerical methods and modern machine learning in the efficient solution of PDEs.

Mechanics-Informed Inverse Design and Optimization for Wrinkle Control in Fiber-Reinforced Bilayers

Xinyu WANG Beijing Normal-Hong Kong Baptist University Peoples Rep of China Co-Author(s):    Xiaoyi Chen, Hao Wang, Peijun Gan, Junlong Dai, Tianyu Shu, Yuxuan Chen, Fanze Yang, Yongtao Lyu

Abstract: Controlling wrinkling in fiber-reinforced bilayers is critical for applications from anti-aging skincare to skin substitutes. While forward predictions are well-developed, efficient optimization and inverse-design strategies targeting microstructural controls---such as fiber pretension and orientation---remain limited. We develop a mechanics-informed framework to tune wrinkle onset and morphology in a skin-inspired bilayer. Using a semi-analytical morphoelastic plate model, we quantify how five design variables ($h, K, \beta, \lambda_f, \phi_f$) govern the critical wrinkling threshold $g_{10}^*$ and mode number $n^*$. This reduced-order formulation enables high-throughput exploration up to $\phi_f=0.85$, providing a computational basis for large-scale optimization challenging for standard 3D finite element simulations. Global sensitivity and mechanical perturbation analyses identify fiber pretension as the dominant regulator of $g_{10}^*$, while $n^*$ is primarily controlled by thickness and orientation. For optimal inverse design, we implement two complementary routes: (i) a regularized mechanics-based optimization leveraging semi-analytical stability conditions, and (ii) a surrogate-assisted search (BPNN--GA) for global scalability. The resulting designs provide physics-consistent tuning strategies for aging skin models, offering quantitative guidance for personalized, skin-inspired material design.

Deep learning methods for singular variational problems with Lavrentiev phenomena

Xianmin XU Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Dong Wang, Boyi Zou

Abstract: Deep learning has gained significant development in the field of scientific computing, especially in its application to solve problems related to differential operators using deep neural networks. However, the utilization of neural networks to solve problems involving singularities still faces challenges. In this talk, we will discuss the failure of deep learning methods for the singular variational problems exhibiting the Lavrentiev phenomenon. For such problems, we show the standard deep Ritz method and some variants fail to detect the singular minimizers. We then introduce a guiding term that renders the neural network to explore solutions as desired during training. Numerical experiments demonstrate that the method achieves much better approximations than the previous methods. Furthermore, we apply the same algorithm to solve problems with regular solutions to show the robustness of the proposed method.

Electrically controlled self-similar evolution of viscous fingering patterns

Meng Zhao Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we investigate the nonlinear dynamics of a moving interface in a Hele-Shaw cell subject to an in-plane applied electric field. We develop a spectrally accurate boundary integral method where a coupled integral equation system is formulated. Our nonlinear results reveal that positive currents restrain finger ramification and promote overall stabilization of patterns. On the other hand, negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region. When no flux is injected, and a negative current is utilized, the interface tends to approach the origin and break up into several drops. We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law.