Abstract--We address the approximation of entropy solutions to initial-boundary value problems for nonlinear strictly hyperbolic conservation laws using neural networks. A general and systematic framework is introduced for the design of efficient and reliable learning algorithms, combining fast convergence during training with accurate predictions. The methodology that relies on solving a certain relaxed related problem is assessed through a series of one-dimensional scalar test cases. These numerical experiments demonstrate the potential of the methodology developed in this paper and its applicability to more complex industrial scenarios. Nonlinear hyperbolic conservation laws play a central role in the mathematical modeling of physical systems where transport and wave propagation phenomena dominate.

The Riemann problem is fundamental in the computational modeling of hyperbolic partial differential equations, enabling the development of stable and accurate upwind schemes. While exact solvers provide robust upwinding fluxes, their high computational cost necessitates approximate solvers. Although approximate solvers achieve accuracy in many scenarios, they produce inaccurate solutions in certain cases. To overcome this limitation, we propose constructing neural network-based surrogate models, trained using supervised learning, designed to map interior and exterior conservative state variables to the corresponding exact flux. Specifically, we propose two distinct approaches: one utilizing a vanilla neural network and the other employing a bi-fidelity neural network. The performance of the proposed approaches is demonstrated through applications to one-dimensional and two-dimensional partial differential equations, showcasing their robustness and accuracy.

Designing re-entry vehicles requires accurate predictions of hypersonic flow around their geometry. Rapid prediction of such flows can revolutionize vehicle design, particularly for morphing geometries. We evaluate advanced neural operator models such as Deep Operator Networks (DeepONet), parameter-conditioned U-Net, Fourier Neural Operator (FNO), and MeshGraphNet, with the objective of addressing the challenge of learning geometry-dependent hypersonic flow fields with limited data. Specifically, we compare the performance of these models for two grid types: uniform Cartesian and irregular grids. To train these models, we use 36 unique elliptic geometries for generating high-fidelity simulations with a high-order entropy-stable DGSEM solver, emphasizing the challenge of working with a scarce dataset. We evaluate and compare the four operator-based models for their efficacy in predicting hypersonic flow field around the elliptic body. Moreover, we develop a novel framework, called Fusion DeepONet, which leverages neural field concepts and generalizes effectively across varying geometries. Despite the scarcity of training data, Fusion DeepONet achieves performance comparable to parameter-conditioned U-Net on uniform grids while it outperforms MeshGraphNet and vanilla DeepONet on irregular, arbitrary grids. Fusion DeepONet requires significantly fewer trainable parameters as compared to U-Net, MeshGraphNet, and FNO, making it computationally efficient. We also analyze the basis functions of the Fusion DeepONet model using Singular Value Decomposition. This analysis reveals that Fusion DeepONet generalizes effectively to unseen solutions and adapts to varying geometries and grid points, demonstrating its robustness in scenarios with limited training data.

We present GoRINNs: numerical analysis-informed (shallow) neural networks for the solution of inverse problems of non-linear systems of conservation laws. GoRINNs is a hybrid/blended machine learning scheme based on high-resolution Godunov schemes for the solution of the Riemann problem in hyperbolic Partial Differential Equations (PDEs). In contrast to other existing machine learning methods that learn the numerical fluxes or just parameters of conservative Finite Volume methods, relying on deep neural networks (that may lead to poor approximations due to the computational complexity involved in their training), GoRINNs learn the closures of the conservation laws per se based on "intelligently" numerical-assisted shallow neural networks. Due to their structure, in particular, GoRINNs provide explainable, conservative schemes, that solve the inverse problem for hyperbolic PDEs, on the basis of approximate Riemann solvers that satisfy the Rankine-Hugoniot condition. The performance of GoRINNs is assessed via four benchmark problems, namely the Burgers', the Shallow Water, the Lighthill-Whitham-Richards and the Payne-Whitham traffic flow models. The solution profiles of these PDEs exhibit shock waves, rarefactions and/or contact discontinuities at finite times. We demonstrate that GoRINNs provide a very high accuracy both in the smooth and discontinuous regions.

Neural operator learning models have emerged as very effective surrogates in data-driven methods for partial differential equations (PDEs) across different applications from computational science and engineering. Such operator learning models not only predict particular instances of a physical or biological system in real-time but also forecast classes of solutions corresponding to a distribution of initial and boundary conditions or forcing terms. % DeepONet is the first neural operator model and has been tested extensively for a broad class of solutions, including Riemann problems. Transformers have not been used in that capacity, and specifically, they have not been tested for solutions of PDEs with low regularity. % In this work, we first establish the theoretical groundwork that transformers possess the universal approximation property as operator learning models. We then apply transformers to forecast solutions of diverse dynamical systems with solutions of finite regularity for a plurality of initial conditions and forcing terms. In particular, we consider three examples: the Izhikevich neuron model, the tempered fractional-order Leaky Integrate-and-Fire (LIF) model, and the one-dimensional Euler equation Riemann problem. For the latter problem, we also compare with variants of DeepONet, and we find that transformers outperform DeepONet in accuracy but they are computationally more expensive.

In this study, we introduce a method based on Separable Physics-Informed Neural Networks (SPINNs) for effectively solving the BGK model of the Boltzmann equation. While the mesh-free nature of PINNs offers significant advantages in handling high-dimensional partial differential equations (PDEs), challenges arise when applying quadrature rules for accurate integral evaluation in the BGK operator, which can compromise the mesh-free benefit and increase computational costs. To address this, we leverage the canonical polyadic decomposition structure of SPINNs and the linear nature of moment calculation, achieving a substantial reduction in computational expense for quadrature rule application. The multi-scale nature of the particle density function poses difficulties in precisely approximating macroscopic moments using neural networks. To improve SPINN training, we introduce the integration of Gaussian functions into SPINNs, coupled with a relative loss approach. This modification enables SPINNs to decay as rapidly as Maxwellian distributions, thereby enhancing the accuracy of macroscopic moment approximations. The relative loss design further ensures that both large and small-scale features are effectively captured by the SPINNs. The efficacy of our approach is demonstrated through a series of five numerical experiments, including the solution to a challenging 3D Riemann problem. These results highlight the potential of our novel method in efficiently and accurately addressing complex challenges in computational physics.

Developing the proper representations for simulating high-speed flows with strong shock waves, rarefactions, and contact discontinuities has been a long-standing question in numerical analysis. Herein, we employ neural operators to solve Riemann problems encountered in compressible flows for extreme pressure jumps (up to $10^{10}$ pressure ratio). In particular, we first consider the DeepONet that we train in a two-stage process, following the recent work of Lee and Shin, wherein the first stage, a basis is extracted from the trunk net, which is orthonormalized and subsequently is used in the second stage in training the branch net. This simple modification of DeepONet has a profound effect on its accuracy, efficiency, and robustness and leads to very accurate solutions to Riemann problems compared to the vanilla version. It also enables us to interpret the results physically as the hierarchical data-driven produced basis reflects all the flow features that would otherwise be introduced using ad hoc feature expansion layers. We also compare the results with another neural operator based on the U-Net for low, intermediate, and very high-pressure ratios that are very accurate for Riemann problems, especially for large pressure ratios, due to their multiscale nature but computationally more expensive. Overall, our study demonstrates that simple neural network architectures, if properly pre-trained, can achieve very accurate solutions of Riemann problems for real-time forecasting.

In this paper, we introduce an improved version of the fifth-order weighted essentially non-oscillatory (WENO) shock-capturing scheme by incorporating deep learning techniques. The established WENO algorithm is improved by training a compact neural network to adjust the smoothness indicators within the WENO scheme. This modification enhances the accuracy of the numerical results, particularly near abrupt shocks. Unlike previous deep learning-based methods, no additional post-processing steps are necessary for maintaining consistency. We demonstrate the superiority of our new approach using several examples from the literature for the two-dimensional Euler equations of gas dynamics. Through intensive study of these test problems, which involve various shocks and rarefaction waves, the new technique is shown to outperform traditional fifth-order WENO schemes, especially in cases where the numerical solutions exhibit excessive diffusion or overshoot around shocks.

In this paper, a modified version of conservative Physics-informed Neural Networks (cPINN for short) is provided to construct the weak solutions of Riemann problem for the hyperbolic scalar conservation laws in non-conservative form. To demonstrate the results, we use the model of generalized Buckley-Leverett equation (GBL equation for short) with discontinuous porosity in porous media. By inventing a new unknown, the GBL equation is transformed into a two-by-two resonant hyperbolic conservation laws in conservative form. The modified method of cPINN is invented to overcome the difficulties due to the discontinuity of the porosity and the appearance of the critical states (near vacuum) in the Riemann data. We experiment with our idea by using a deep learning algorithm to solve the GBL equation in both conservative and non-conservative forms, as well as the cases of critical and non-critical states. This method provides a combination of two different neural networks and corresponding loss functions, one is for the two-by-two resonant hyperbolic system, and the other is for the scalar conservation law with a discontinuous perturbation term in the non-convex flux. The technique of re-scaling to the unknowns is adopted to avoid the oscillation of the Riemann solutions in the cases of critical Riemann data. The solutions constructed by the modified cPINN match the exact solutions constructed by the theoretical analysis for hyperbolic conservation laws. In addition, the solutions are identical in both conservative and non-conservative cases. Finally, we compare the performance of the modified cPINN with numerical method called WENO5. Whereas WENO5 struggles with the highly oscillation of approximate solutions for the Riemann problems of GBL equation in non-conservative form, cPINN works admirably.

A least-squares neural network (LSNN) method was introduced for solving scalar linear and nonlinear hyperbolic conservation laws (HCLs) in [7, 6]. This method is based on an equivalent least-squares (LS) formulation and uses ReLU neural network as approximating functions, making it ideal for approximating discontinuous functions with unknown interface location. In the design of the LSNN method for HCLs, the numerical approximation of differential operators is a critical factor, and standard numerical or automatic differentiation along coordinate directions can often lead to a failed NN-based method. To overcome this challenge, this paper rewrites HCLs in their divergence form of space and time and introduces a new discrete divergence operator. As a result, the proposed LSNN method is free of penalization of artificial viscosity. Theoretically, the accuracy of the discrete divergence operator is estimated even for discontinuous solutions. Numerically, the LSNN method with the new discrete divergence operator was tested for several benchmark problems with both convex and non-convex fluxes, and was able to compute the correct physical solution for problems with rarefaction, shock or compound waves. The method is capable of capturing the shock of the underlying problem without oscillation or smearing, even without any penalization of the entropy condition, total variation, and/or artificial viscosity.