Physics-informed neural networks (PINNs) have been widely applied in different fields due to their effectiveness in solving partial differential equations (PDEs). However, the accuracy and efficiency of PINNs need to be considerably improved for scientific and commercial purposes. To address this issue, we systematically propose a novel dimension-augmented physics-informed neural network (DaPINN), which simultaneously and significantly improves the accuracy and efficiency of the base PINN. In the DaPINN model, we manipulate the dimensionality of the network input by inserting additional sample features and then incorporate the expanded dimensionality into the loss function. Moreover, we verify the effectiveness of power series augmentation, Fourier series augmentation and replica augmentation in both forward and backward problems. In most experiments, the error of DaPINN is 1 ∼2 orders of magnitude lower than that of the base PINN. The results show that the DaPINN outperforms the original PINN in terms of both accuracy and efficiency with a reduced dependence on the number of sample points. We also discuss the computational complexity of the DaPINN, its network size implications, other implementations of the DaPINN and the compatibility of DaPINN's methods with residual-based adaptive refinement (RAR), self-adaptive physics-informed neural networks (SA-PINNs) and gradient-enhanced physics-informed neural networks (gPINNs).

This work introduces an ‘equation-free’ non-intrusive model order reduction (NIMOR) method for surrogate modeling of time-domain electromagnetic wave propagation. The nested proper orthogonal decomposition (POD) method, as a prior dimensionality reduction technique, is employed to extract the time- and parameter-independent reduced basis (RB) functions from a collection of high-fidelity (HF) solutions (or snapshots) on a properly chosen training parameter set. A dynamic mode decomposition (DMD) method, resulting in a further dimension reduction of the NIMOR method, is then used to predict the reduced-order coefficient vectors for future time instants on the previous training parameter set. The radial basis function (RBF) is employed for approximating the reduced-order coefficient vectors at a given untrained parameter in the future time instants, leading to the applicability of DMD method to parameterized problems. A main advantage of the proposed method is the use of a multi-step procedure consisting of the POD, DMD and RBF techniques to accurately and quickly recover field solutions from a few large-scale simulations. Numerical experiments for the scattering of a plane wave by a dielectric disk, by a multi-layer disk, and by a 3-D dielectric sphere nicely illustrate the performance of the NIMOR method.

Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications. Alternative to traditional numerical schemes, learning-based solvers utilize the representation power of DNNs to approximate the input-output relations in an automated manner. However, the lack of physics-in-the-loop often makes it difficult to construct a neural network solver that simultaneously achieves high accuracy, low computational burden, and interpretability. In this work, focusing on a class of evolutionary PDEs characterized by decomposable operators, we show that the classical “operator splitting” technique can be adapted to design neural network architectures. This gives rise to a learning-based PDE solver, which we name Deep Operator-Splitting Network (DOSnet). Such non-black-box network design is constructed from the physical rules and operators governing the underlying dynamics, and is more efficient and flexible than the classical numerical schemes and standard DNNs. To demonstrate the advantages of our new AI-enhanced PDE solver, we train and validate it on several types of operator-decomposable differential equations. We also apply DOSnet to nonlinear Schrödinger equations which have important applications in the signal processing for modern optical fiber transmission systems, and experimental results show that our model has better accuracy and lower computational complexity than numerical schemes and the baseline DNNs.

This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by formulating the problem of finding a reproducing kernel Hilbert space (RKHS) to approximate a PDE solution. The approximated solution lies in the span of base functions generated by evaluating derivatives of different orders of kernels at sample points. However, the RKHS specified by GPs can result in an expensive computational burden due to the cubic computation order of the matrix inverse. Therefore, we conjecture that a solution exists on a “condensed” subspace that can achieve similar approximation performance, and we propose a SGP-based method to reformulate the optimization problem in the “condensed” subspace. This significantly reduces the computation burden while retaining desirable accuracy. The paper rigorously formulates this problem and provides error analysis and numerical experiments to demonstrate the effectiveness of this method. The numerical experiments show that the SGP method uses fewer than half the uniform samples as inducing points and achieves comparable accuracy to the GP method using the same number of uniform samples, resulting in a significant reduction in computational cost.Our contributions include formulating the nonlinear PDE problem as an optimization problem on a “condensed” subspace of RKHS using SGP, as well as providing an existence proof and rigorous error analysis. Furthermore, our method can be viewed as an extension of the GP method to account for general positive semi-definite kernels.

We present a Calderón preconditioning scheme for the symmetric formulation of the forward electroencephalographic (EEG) problem that cures both the dense-discretization and the high-contrast breakdown. Unlike existing Calderón schemes presented for the EEG problem, it is refinement-free, that is, the electrostatic integral operators are not discretized with basis functions defined on the barycentrically-refined dual mesh. In fact, in the preconditioner, we reuse the original system matrix thus reducing computational burden. Moreover, the proposed formulation gives rise to a symmetric, positive-definite system of linear equations, which allows the application of the conjugate gradient method, an iterative method that exhibits a smaller computational cost compared to other Krylov subspace methods applicable to non-symmetric problems. Numerical results corroborate the theoretical analysis and attest of the efficacy of the proposed preconditioning technique on both canonical and realistic scenarios.

In this short note, we discuss the circumstances that can lead to a failure to observe the design order of discretization error convergence in accuracy verification when solving a time-dependent problem. In particular, we discuss the problem of failing to observe the design order of spatial accuracy with extremely small time steps. This can cause a serious problem because then one would wind up trying to find a coding error that does not exist. This short note clarifies the mechanism causing this failure to observe a design order of discretization error convergence in accuracy verification when solving time-dependent problems, and provides a guide for avoiding such pitfalls.

Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies are often heuristic and rely on domain-specific knowledge or trial-and-error. We treat the process of adaptive mesh refinement as a local, sequential decision-making problem under incomplete information, formulating AMR as a partially observable Markov decision process. Using a deep reinforcement learning (RL) approach, we train policy networks for AMR strategy directly from numerical simulation. The training process does not require an exact solution or a high-fidelity ground truth to the partial differential equation (PDE) at hand, nor does it require a pre-computed training dataset. The local nature of our deep RL (DRL) allows the policy network to be trained inexpensively on much smaller problems than those on which they are deployed. The new DRL-AMR method is not specific to any particular PDE, problem dimension, or numerical discretization. The RL policy networks, trained on simple examples, can generalize to more complex problems, and can flexibly incorporate diverse problem physics. To that end, we apply the method to a range of PDEs, using a variety of high-order discontinuous Galerkin and hybridizable discontinuous Galerkin finite element discretizations. We show that the resultant DRL policies are competitive with common AMR heuristics and strike a favorable balance between accuracy and cost such that they often lead to a higher accuracy per problem degree of freedom, and are effective across a wide class of PDEs and problems.

Based on the Rayleigh bubble dynamics equation a mass transfer model for cavitation of binary alkane mixtures is presented. Raoult's and Dalton's law, simple mixing rules, and an accurate Equation of State are utilized. The model is implemented into an in-house CFD code. For solver validation pure species literature cases are taken. The method is applied to a lighter n-octane/n-heptane and a heavier n-dodecane/n-heptane mixture in a rarefaction tube and a hydrofoil test case. Segregation of the species is observed during cavitation due to their different mass transfer rates. While for the lighter mixture, mass transfer of both species only moderately deviates, a significantly higher mass transfer of n-heptane compared to n-dodecane is observed for the heavier mixture, where the saturation pressure differs two orders of magnitude between the mixture ingredients. The strong segregation of the heavier mixture is associated with a predominant amount of n-heptane in the vapor phase. As a consequence, vapor composition is strongly affected by the volatilities of mixture ingredients.

The opacity of FRTE depends on not only the material temperature but also the frequency, whose values may vary several orders of magnitude for different frequencies. The gray radiation diffusion and frequency-dependent diffusion equations are two simplified models that can approximate the solution to FRTE in the thick opacity regime. The frequency discretization for the two limit models highly affects the numerical accuracy. However, classical frequency discretization for FRTE considers only the absorbing coefficient. In this paper, we propose a new decomposed multi-group method for frequency discretization that is not only AP in both gray radiation diffusion and frequency-dependent diffusion limits, but also the frequency discretization of the limiting models can be tuned. Based on the decomposed multi-group method, a full AP scheme in frequency, time, and space is proposed. Several numerical examples are used to verify the performance of the proposed scheme.

In the analysis of magnetohydrodynamic (MHD) flow, the Lorentz force significantly affects energy properties because the work generated by the Lorentz force changes the kinetic and magnetic energies. Therefore, the Lorentz force and energy conversion should be predicted accurately. Some energy conservation schemes have been proposed and validated. However, the influences of the Lorentz force discretization on conservation and conversion of energy have not yet been clarified. In this study, a conservative finite difference method is constructed for incompressible MHD flows considering the induced magnetic field. We compare the difference in energy conservation properties among three methods of calculating the Lorentz force. The Lorentz forces are calculated in conservative and non-conservative forms, and both compact and wide-range interpolations of magnetic flux density are used to calculate the non-conservative Lorentz force. The compact interpolation method proposed in this study can perform conversions between conservative and non-conservative forms of the Lorentz force even when using the finite difference method. The present numerical method improves the conservation of transport quantity. Five models were analyzed, and the accuracy and convergence of the present numerical method were verified. From the viewpoint of the conservation of the total energy in an ideal inviscid periodic MHD flow, we consider that the calculation using compact interpolation for the Lorentz force is appropriate. This method preserves the total energy even on non-uniform grids. Moreover, the divergence-free condition of the magnetic flux density is discretely satisfied even without the correction of the magnetic flux density. The present numerical method can capture the Hartmann layer in the propagation of an Alfvén wave and accurately predict the tendency of energy attenuation in the analysis of a Taylor decaying vortex under magnetic fields. Analysis of the Orszag–Tang vortex reveals energy dissipation processes and the generation of high current densities. The present numerical method has excellent energy conservation properties and can accurately predict energy conversion. Therefore, this method can contribute to understanding complex unsteady MHD flows.

Ha et al. [1] have constructed a novel set of smoothness indicators by using the L1-norm measure and introducing a user-tunable parameter to propose a new WENO scheme, dubbed WENO-NS. Kim et al. [2] have improved WENO-NS with another user-tunable parameter to balance the contribution of substencils, leading to the WENO-P scheme. Recently, Rathan et al. [3] have modified WENO-P by devising a higher-order global smoothness indicator. The associated MWENO-P scheme aimed to fix the issue of WENO-P and WENO-NS that they cannot achieve the optimal convergence orders at the critical points where the first and second derivatives are zero. The above-said schemes outperform the existing many WENO schemes in some senses. However, their smoothness indicators violate the symmetry-preserving property preserved by those of many other well-established WENO schemes and rely on one or two user-tunable parameters. In the present study, a new set of smoothness indicators satisfying the symmetry-preserving property without using any user-tunable parameters is constructed. The resultant scheme is called the improved MWENO-P scheme and abbreviated as IMWENO-P. We provide detailed theoretical analysis and numerical examples to verify that the new scheme can attain optimal convergence orders even in the presence of second-order critical points. Euler equations are simulated to demonstrate the enhancements of the new scheme, such as the lower dissipation and better resolution.

We propose a multi-phase, multi-resolution SPH method for fluid/solid interaction with multi-GPU implementation and dynamic load balancing following the movement of the refinement regions. The primary design goal of this framework is to maintain the efficiency of the single-resolution SPH model running on a single GPU. To this end, a multi-background mesh is introduced, and the domain is regarded as a nested multi-domain with different resolutions. Validation using both a δ-SPH and Riemann SPH model is shown, and applications to the simulation of the water entry of a projectile with a high Froude number are considered, with comparisons to experimental data from three challenging test cases, showing the proposed model's ability to correctly reproduce the free surface evolution on water entry, the motion of the projectile, and the formation and evolution of multiple cavities depending on entry angle and velocity. An analysis of the computational performance and resolutions achieved (up to 120 million particles) is also provided across several test cases.

In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of the source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions.

Abstract not available

We consider the reliable implementation of high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems. We construct a reliable algorithm to merge small interface elements with their surrounding elements to automatically generate the finite element mesh whose elements are large with respect to both domains. We propose new basis functions for the interface elements to control the growth of the condition number of the stiffness matrix in terms of the finite element approximation order, the number of elements of the mesh, and the interface deviation which quantifies the mesh resolution of the geometry of the interface. Numerical examples are presented to illustrate the competitive performance of the method.

We present a 3D adaptive octree based numerical method for the simulation of binary solidification, with the temperature and the concentration fields being strongly coupled at the solidification front. The volume of fluid approach (VOF) is used to advance the front, which is geometrically reconstructed as a sharp interface separating the liquid and the solid domains. On basis of this, the embedded boundary method (EBM), together with the finite volume method, are employed to discretize the temperature and concentration fields at two sides of the interface in a sharp manner. The major novelty of the present study are displayed in several aspects: It is the first attempt to develop fully sharp schemes for the simulation of binary solidification in the VOF framework, while previous VOF-type methods consider the jump conditions at the interface as source terms; second the geometrically reconstructed interface and the EBM enable us to capture those discontinuities accurately without any artificial smearing, and particularly, we will show how to construct a second-order scheme for the flux jump condition of the temperature field across the interface; finally compared to other sharp schemes, we will show that the present method has natural advantages to adapt with the fluid flow solvers, because the hybrid VOF-EBM method guarantees good performance in solving the complex solid-fluid coupling problems based on the finite volume framework. Owing to all these advantages, we will show that regardless of pure substance or binary solution, all the numerical results agree well with the benchmark results. In particular, we apply the numerical method to model the solidification of a Ni-Cu alloy, which is thought to be very challenging owing to the high Lewis number. Besides, coupling to the fluid flow solver, we demonstrate that this method can predict the solidification/melting of an ice cylinder/sphere accurately in a salted solution under forced convection, by which the fluid, temperature and concentration equations are all solved sharply.

The hypersonic flow is in a thermochemical nonequilibrium state due to the high-temperature caused by the strong shock compression. In a thermochemical nonequilibrium flow, the distribution of molecular internal energy levels strongly deviates from the equilibrium distribution (i.e., the Boltzmann distribution). It is intractable to directly obtain the microscopic nonequilibrium distribution from existed experimental measurements usually described by macroscopic field variables such as temperature or velocity. Motivated by the idea of deep multi-scale multi-physics neural network (DeepMMNet) proposed in [1], we develop in this paper a data assimilation framework called DeepStSNet to accurately reconstruct the quantum state-resolved thermochemical nonequilibrium flowfield by using sparse experimental measurements of vibrational temperature and pre-trained deep neural operator networks (DeepONets). In particular, we first construct several DeepONets to express the coupled dynamics between field variables in the thermochemical nonequilibrium flow and to approximate the state-to-state (StS) approach, which traces the variation of each vibrational level of molecule accurately. These proposed DeepONets are then trained by using the numerical simulation data, and would later be served as building blocks for the DeepStSNet. We demonstrate the effectiveness and accuracy of DeepONets with different test cases showing that the density and energy of vibrational groups as well as the temperature and velocity fields are predicted with high accuracy. We then extend the architectures of DeepMMNet by considering a simplified thermochemical nonequilibrium model, i.e., the 2T model, showing that the entire thermochemical nonequilibrium flowfield is well predicted by using scattered measurements of full or even partial field variables. We next consider a more accurate and complex thermochemical nonequilibrium model, i.e., the StS-CGM model, and develop a DeepStSNet for this model. In this case, we employ the coarse-grained method, which divides the vibrational levels into groups (vibrational bins), to alleviate the computational cost for the StS approach in order to achieve a fast but reliable prediction with DeepStSNet. We test the present DeepStSNet framework with sparse numerical simulation data showing that the predictions are in excellent agreement with the reference data for test cases. We further employ the DeepStSNet to assimilate a few experimental measurements of vibrational temperature obtained from the shock tube experiment, and the detailed non-Boltzmann vibrational distribution of molecule oxygen is reconstructed by using the sparse experimental data for the first time. Moreover, by considering the inevitable uncertainty in the experimental data, an average strategy in the predicting procedure is proposed to obtain the most probable predicted fields. The present DeepStSNet is general and robust, and can be applied to build a bridge from sparse measurements of macroscopic field variables to a microscopic quantum state-resolved flowfield. This kind of reconstruction is beneficial for exploiting the experimental measurements and uncovering the hidden physicochemical processes in hypersonic flows.

The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability.The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Numerical analysis of the new basis functions reveals that their higher dual regularity does not only lead to conservation, but also yields an L2-stable basis for the combination technique. Accordingly, further numerical experiments applying the combination technique to a semi-Lagrangian Vlasov–Poisson solver in six dimensions show a stabilizing effect of the biorthogonal and full weighting bases on the simulations.

Boundary-optimized summation-by-parts (SBP) finite difference operators for second derivatives with variable coefficients are presented. The operators achieve increased accuracy by utilizing non-equispaced grid points close to the boundaries of the grid. Using the optimized operators we formulate SBP schemes for the acoustic and elastic operators defined directly on curvilinear multiblock domains. Numerical studies of the acoustic and elastic wave equations demonstrate that, compared to traditional SBP difference operators, the new operators provide increased accuracy for surface waves as well as block interfaces in multiblock grids. For instance, simulations of Rayleigh waves demonstrate that the boundary-optimized operators more than halve the runtime required for a given error tolerance.

The discrete-equations method (DEM) [1] provides a universal approach to solve multi-phase-flow equations as it combines the solutions of pairwise Riemann problems. Although very robust, the original DEM with piecewise-constant volume fractions suffers from strong diffusion preventing accurate interface capturing. High-order interface reconstruction, however, introduces a restrictive time-step limit.This paper presents RDEMIC, a robust extension of DEM for accurate interface capturing on Cartesian meshes. By a modified partitioning of the Riemann solutions and a specific combination of fluxes and non-conservative terms, the time-step restriction is effectively prevented, which is critical for making the method practically applicable. Moreover, the accuracy of interface and shock-wave propagation is maintained. RDEMIC is not limited to two-phase flow but defined for an arbitrary number of phases.The method is combined with a THINC scheme [2] to reconstruct volume fractions. The reconstruction is enhanced by a positivity-preserving averaging procedure, which is consistent with the underlying multi-stage Runge–Kutta scheme of the flow solver.The resulting scheme consisting of RDEMIC and the positivity-preserving THINC reconstruction is very robust and captures the interface with high accuracy. We demonstrate its performance for various cases of shock-interface interactions, which show very good agreement with reference results from literature.