Changxin Qiu

Wei-Wei Han, Fukeng Huang, Changxin Qiu

We develop and analyze high-order ensemble schemes for the unsteady Navier--Stokes--Darcy system with uncertain initial conditions, forcing terms, hydraulic conductivity tensors, and Lions-Beavers-Joseph-Saffman interface conditions. The proposed schemes which are called GSAV-GBDF$k$-Ensemble schemes integrate a partitioned decoupling strategy, the generalized scalar auxiliary variable (GSAV) approach, and generalized BDF$k$ discretizations. This framework achieves high-order temporal accuracy and long-time stability, permits explicit treatment of the nonlinear term, and facilitates an efficient ensemble implementation for multiple parameter realizations by sharing a single, unified coefficient matrix at each time step. Moreover, the numerical solutions are shown to satisfy uniform-in-time bounds without time-step restrictions. Owing to the ensemble formulation, the resulting linear systems share common coefficient matrices, which significantly improves computational efficiency. We further establish optimal-order error estimates for the proposed high-order schemes. Numerical results are included to confirm the theoretical analysis and to illustrate the accuracy, stability, and efficiency of the proposed methods.

Mengjia Chen, Changxin Qiu, Zhiping Mao et al.

The numerical simulation of interaction between free flow and porous media, governed by coupled Stokes/Navier--Stokes--Darcy flows, is critical for understanding fluid filtration and physiological transport, yet it is hindered by the high computational cost of resolving interface heterogeneities and the instability of long-term predictions. While deep learning offers surrogate modeling potential, existing frameworks often suffer from exponential error accumulation and poor convergence in multi-physics regimes. To address these limitations, we propose ViT-K, a novel few-shot learning model designed to learn the spatiotemporal evolution of coupled flows from sparse datasets. The ViT-K framework effectively reconstructs the global flow physics on a low-dimensional manifold by combining Vision Transformers (ViT) to capture heterogeneous interfacial features with the Koopman operator to linearize temporal dynamics. By lifting nonlinear dynamics into a globally linear observable space, the ViT-K model provides stability by design, ensuring that prediction errors grow linearly rather than exponentially over time. This theoretical property enables reliable long-term extrapolation even in small-sample regimes. Numerical experiments on benchmark coupled systems demonstrate that ViT-K not only captures complex interface physics with high fidelity but also exhibits exceptional robustness against measurement noise by acting as an implicit spectral filter. The proposed method significantly outperforms traditional solvers in inference speed while maintaining physical consistency, offering a robust paradigm for real-time multiphysics forecasting.

Changxin Qiu, Jue Yan

Motivated by finite volume scheme, a cell-average based neural network method is proposed. The method is based on the integral or weak formulation of partial differential equations. A simple feed forward network is forced to learn the solution average evolution between two neighboring time steps. Offline supervised training is carried out to obtain the optimal network parameter set, which uniquely identifies one finite volume like neural network method. Once well trained, the network method is implemented as a finite volume scheme, thus is mesh dependent. Different to traditional numerical methods, our method can be relieved from the explicit scheme CFL restriction and can adapt to any time step size for solution evolution. For Heat equation, first order of convergence is observed and the errors are related to the spatial mesh size but are observed independent of the mesh size in time. The cell-average based neural network method can sharply evolve contact discontinuity with almost zero numerical diffusion introduced. Shock and rarefaction waves are well captured for nonlinear hyperbolic conservation laws.

Changxin Qiu, Aaron Bendickson, Joshua Kalyanapu et al.

In this paper we consider utilizing a residual neural network (ResNet) to solve ordinary differential equations. Stochastic gradient descent method is applied to obtain the optimal parameter set of weights and biases of the network. We apply forward Euler, Runge-Kutta2 and Runge-Kutta4 finite difference methods to generate three sets of targets training the ResNet and carry out the target study. The well trained ResNet behaves just as its counterpart of the corresponding one-step finite difference method. In particular, we carry out (1) the architecture study in terms of number of hidden layers and neurons per layer to find the optimal ResNet structure; (2) the target study to verify the ResNet solver behaves as accurate as its finite difference method counterpart; (3) solution trajectory simulation. Even the ResNet solver looks like and is implemented in a way similar to forward Euler scheme, its accuracy can be as high as any one step method. A sequence of numerical examples are presented to demonstrate the performance of the ResNet solver.