Discovering underlying partial differential equations (PDEs) from observational data has important implications across fields. It bridges the gap between theory and observation, enhancing our understanding of complex systems in applications. In this paper, we propose a novel approach, termed physics-informed sparse optimization (PIS), for learning surface PDEs. Our approach incorporates both $L_2$ physics-informed model loss and $L_1$ regularization penalty terms in the loss function, enabling the identification of specific physical terms within the surface PDEs. The unknown function and the differential operators on surfaces are approximated by some extrinsic meshless methods. We provide practical demonstrations of the algorithms including linear and nonlinear systems. The numerical experiments on spheres and various other surfaces demonstrate the effectiveness of the proposed approach in simultaneously achieving precise solution prediction and identification of unknown PDEs.

The reduction of Hamiltonian systems aims to build smaller reduced models, valid over a certain range of time and parameters, in order to reduce computing time. By maintaining the Hamiltonian structure in the reduced model, certain long-term stability properties can be preserved. In this paper, we propose a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks. Their training is coupled in order to learn the encoder-decoder operators and the reduced dynamics simultaneously. Several test cases on non-linear wave dynamics show that the method has better reduction properties than standard linear Hamiltonian reduction methods.

A deep learning-based computational method is proposed for soft matter dynamics – the deep Onsager-Machlup method (DOMM). It combines the brute forces of deep neural networks (DNNs) with the fundamental physics principle – Onsager-Machlup variational principle (OMVP). In the DOMM, the trial solution to the dynamics is constructed by DNNs that allow us to explore a rich and complex set of admissible functions. It outperforms the Ritz-type variational method where one has to impose carefully-chosen trial functions. This capability endows the DOMM with the potential to solve rather complex problems in soft matter dynamics that involve multiple physics with multiple slow variables, multiple scales, and multiple dissipative processes. Actually, the DOMM can be regarded as an extension of the deep Ritz method (DRM) developed by E and Yu that uses DNNs to solve static problems in physics. In this work, as the first step, we focus on the validation of the DOMM as a useful computational method by using it to solve several typical soft matter dynamic problems: particle diffusion in dilute solutions, and two-phase dynamics with and without hydrodynamics. The predicted results agree very well with the analytical solution or numerical solution from traditional computational methods. These results show the accuracy and convergence of DOMM and justify it as an alternative computational method for solving soft matter dynamics.

An implicit discrete unified gas kinetic scheme (DUGKS) is developed for multiscale steady flows of binary gas mixtures by solving the Andries-Aoki-Perthame kinetic model (AAP). To ensure the high convergence efficiency for all flow regimes, the microscopic and macroscopic asynchronous iterative strategies are used, where both the macroscopic and microscopic equations are solved iteratively by the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) method. The macroscopic iteration is conducted to solve the macroscopic governing equations containing source terms as an implicit prediction step to evaluate the local equilibrium state of the microscopic evolution, and the macroscopic flux used in the macroscopic iteration is obtained by taking moments of the distribution function. Besides, to keep the asymptotic preserving properties, the numerical flux across the cell interface is reconstructed by the characteristic solution of the kinetic governing equations for both species like the explicit DUGKS for a single gas. Several numerical tests, including the Couette flow, the lid-driven cavity flow, and the flows through a slit of different mixtures, are simulated to verify the accuracy and efficiency of the present scheme for binary mixtures. Furthermore, compared to the explicit DUGKS, the implicit scheme improves the computational efficiency by 1-2 orders of magnitude.

Traditionally, classical numerical schemes have been employed to solve partial differential equations (PDEs) using computational methods. Recently, neural network-based methods have emerged. Despite these advancements, neural network-based methods, such as physics-informed neural networks (PINNs) and neural operators, exhibit deficiencies in robustness and generalization. To address these issues, numerous studies have integrated classical numerical frameworks with machine learning techniques, incorporating neural networks into parts of traditional numerical methods. In this study, we focus on hyperbolic conservation laws by replacing traditional numerical fluxes with neural operators. To this end, we developed loss functions inspired by established numerical schemes related to conservation laws and approximated numerical fluxes using Fourier neural operators (FNOs). Our experiments demonstrated that our approach combines the strengths of both traditional numerical schemes and FNOs, outperforming standard FNO methods in several respects. For instance, we demonstrate that our method is robust, has resolution invariance, and is feasible as a data-driven method. In particular, our method can make continuous predictions over time and exhibits superior generalization capabilities with out-of-distribution (OOD) samples, which are challenges that existing neural operator methods encounter.

It is of great interest to solve the inverse problem of stationary radiative transport equation (RTE) in optical tomography. The standard way is to formulate the inverse problem into an optimization problem, but the bottleneck is that one has to solve the forward problem repeatedly, which is time-consuming. Due to the optical property of biological tissue, in real applications, optical thin and thick regions coexist and are adjacent to each other, and the geometry can be complex. To use coarse meshes and save the computational cost, the forward solver has to be asymptotic preserving across the interface (APAL). In this paper, we propose an offline/online solver for RTE. The cost at the offline stage is comparable to classical methods, while the cost at the online stage is much lower. Two cases are considered. One is to solve the RTE with fixed scattering and absorption cross sections while the boundary conditions vary; the other is when cross sections vary in a small domain and the boundary conditions change many times. The solver can be decomposed into offline/online stages in these two cases. One only needs to calculate the offline stage once and update the online stage when the parameters vary. Our proposed solver is much cheaper when one needs to solve RTE with multiple right-hand sides or when the cross sections vary in a small domain, thus can accelerate the speed of solving inverse RTE problems. We illustrate the online/offline decomposition based on the Tailored Finite Point Method (TFPM), which is APAL on general quadrilateral meshes.

In this paper, we introduce a new stabilized continuous linear element method for solving biharmonic problems. Leveraging the gradient recovery operator, we reconstruct the discrete Hessian for piecewise continuous linear functions. By adding a stability term to the discrete bilinear form, we bypass the need for the discrete Poincaré inequality. We employ Nitsche's method for weakly enforcing boundary conditions. We establish well-posedness of the solution and derive optimal error estimates in energy and $L^2$ norms. Numerical results are provided to validate our theoretical findings.

Whether there is a similarity between two physical processes in the movement of objects and the complexity of behavior is an essential problem in science. How to seek similarity through the adoption of quantitative and qualitative research techniques still remains an urgent challenge we face. To this end, the concepts of similarity transformation matrix and similarity degree are innovatively introduced to describe similarity of orbits between two complicated discrete dynamical systems that seem to be irrelevant. Furthermore, we present a general optimal principle, to give a strict characterization from the perspective of dynamical systems combined with optimization theory. For well-known examples of chaotic dynamical systems, such as Lorenz attractor, Chua’s circuit, Rössler attractor, Chen attractor, Lü attractor and hybrid system, with using of the homotopy idea, some numerical simulation results reveal that a similarity can be found in rich characteristics and complex behaviors of chaotic dynamics via the optimal principle we presented.

Lattice Boltzmann method is a popular approach in computational fluid dynamics, and it can be used explicitly or implicitly. The explicit methods require small time step size which is not desirable. In the fully implicit case, existing approaches either lack a scalable and robust parallel nonlinear solver, or don’t allow the mesh to be fully unstructured preventing the method to be used for the simulation of fluid flows in large domains with complex geometry. In this paper, a parallel fully implicit second-order finite volume lattice Boltzmann method for incompressible flows on unstructured grids is introduced. The lattice Boltzmann equation is discretized by a finite volume method in space and an implicit backward Euler scheme in time. The resulting large sparse nonlinear system of algebraic equations is solved by a highly parallel Schwarz type domain decomposition preconditioned Newton-Krylov algorithm. The proposed method is validated by three benchmark problems with a wide range of Reynolds number: (a) pressure driven Poiseuille flow, (b) lid-driven cavity flows, and (c) viscous flows passing a circular cylinder. The numerical results show that the proposed method is robust for all the test cases and a superlinear speedup is obtained for solving a problem with over ten million degree of freedoms using thousands of processor cores.

Dynamic mode decomposition (DMD), as a data-driven method, has been frequently used to construct reduced-order models (ROMs) due to its good performance in time extrapolation. However, existing DMD-based ROMs suffer from high storage and computational costs for high-dimensional problems. To mitigate this problem, we develop a new DMD-based ROM, i.e., TDMD-GPR, by combining tensor train decomposition (TTD) and Gaussian process regression (GPR), where TTD is used to decompose the high-dimensional tensor into multiple factors, including parameter-dependent and time-dependent factors. Parameter-dependent factor is fed into GPR to build the map between parameter value and factor vector. For any parameter value, multiplying the corresponding parameter-dependent factor vector and the time-dependent factor matrix, the result describes the temporal behavior of the spatial basis for this parameter value and is then used to train the DMD model. In addition, incremental singular value decomposition is adopted to acquire a collection of important instants, which can further reduce the computational and storage costs of TDMD-GPR. The comparison TDMD and standard DMD in terms of computational and storage complexities shows that TDMD is more advantageous. The performance of the TDMD and TDMD-GPR is assessed through several cases, and the numerical results confirm the effectiveness of them.