We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods. Key words. Deep neural network, residual network, recurrent neural network, governing equation discovery 1. Introduction. Recently there has been a growing interest in discovering governing equations numerically using observational data. Earlier efforts include methods using symbolic regression ([5, 43]), equation-free modeling [24], heterogeneous multiscale method (HMM) ([15]), artificial neural networks ([19]), nonlinear regression ([50]), empirical dynamic modeling ([46, 53]), nonlinear Laplacian spectral analysis ([18]), automated inference of dynamics ([44, 12, 13]), etc. More recent efforts start to cast the problem into a function approximation problem, where the unknown governing equations are treated as target functions relating the data for the state variables and their time derivatives. The majority of the methods employ certain sparsitypromoting algorithms to create parsimonious models from a large set of dictionary for all possible models, so that the true dynamics could be recovered exactly ([47]). Many studies have been conducted to effectively deal with noises in data ([7, 40]), corruptions in data ([48]), partial differential equations [38,41], etc. Methods have also been developed in conjunction with model selection approach ([28]), Koopman theory ([6]), and Gaussian process regression ([35]), to name a few. A more recent work resorts to the more traditional means of approximation by using orthogonal polynomials ([52]). The approach seeks accurate numerical approximation to the underlying governing equations, instead of their exact recovery. By doing so, many existing results in polynomial approximation theory can be applied, particularly those on sampling strategies. It was shown in [52] that data from a large number of short bursts of trajectories are more effective for equation recovery than those from a single long trajectory.On the other hand, artificial neural network (ANN), and particularly deep neural network (DNN), has seen tremendous successes in many different disciplines. The number of publications is too large to mention. Here we cite only a few relati...

The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrich splitting, the WENO reconstruction, the physicalconstraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations [21]. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit formulas of the primitive variables and the flux vectors with respect to the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector instead of the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case.Several one-and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.

The density and pressure are positive physical quantities in magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP) numerical schemes for ideal compressible MHD is highly desirable, but remains a challenge especially in the multidimensional cases. In this paper, we first develop uniformly high-order discontinuous Galerkin (DG) schemes which provably preserve the positivity of density and pressure for multidimensional ideal MHD. The schemes are constructed by using the locally divergence-free DG schemes for the symmetrizable ideal MHD equations as the base schemes, a PP limiter to enforce the positivity of the DG solutions, and the strong stability preserving methods for time discretization. The significant innovation is that we discover and rigorously prove the PP property of the proposed DG schemes by using a novel equivalent form of the admissible state set and very technical estimates. Several two-dimensional numerical examples further confirm the PP property, and demonstrate the accuracy, effectiveness and robustness of the proposed PP methods.

This paper first studies the admissible state set G of relativistic magnetohydrodynamics (RMHD). It paves a way for developing physical-constraints-preserving (PCP) schemes for the RMHD equations with the solutions in G. To overcome the difficulties arising from the extremely strong nonlinearities and no explicit formulas of the primitive variables and the flux vectors with respect to the conservative vector, two equivalent forms of G with explicit constraints on the conservative vector are skillfully discovered. The first is derived by analyzing roots of several polynomials and transferring successively them, and further used to prove the convexity of G with the aid of semi-positive definiteness of the second fundamental form of a hypersurface. While the second is derived based on the convexity, and then used to show the orthogonal invariance of G. The Lax-Friedrichs (LxF) splitting property does not hold generally for the nonzero magnetic field, but by a constructive inequality and pivotal techniques, we discover the generalized LxF splitting properties, combining the convex combination of some LxF splitting terms with a discrete divergence-free condition of the magnetic field.Based on the above analyses, several one-and two-dimensional PCP schemes are then studied. In the 1D case, a first-order accurate LxF type scheme is first proved to be PCP under the Courant-Friedrichs-Lewy (CFL) condition, and then the high-order accurate PCP schemes are proposed via a PCP limiter. In the 2D case, the discrete divergence-free condition and PCP property are analyzed for a first-order accurate LxF type scheme, and two sufficient conditions are derived for high-order accurate PCP schemes. Our analysis reveals in theory for the first time that the discrete divergence-free condition is closely connected with the PCP property. Several numerical examples demonstrate the theoretical findings and the performance of numerical schemes.

Numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. This paper presents the first rigorous PP analysis of conservative schemes with the Lax-Friedrichs (LF) flux for one-and multi-dimensional ideal MHD. The significant innovation is the discovery of the theoretical connection between the PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives of the usually-expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by the DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality, which is skillfully constructed by technical estimates. Rigorous PP analysis is then presented for finite volume and discontinuous Galerkin schemes with the LF flux on uniform Cartesian meshes. In the 1D case, the PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In the 2D case, we show that the DDF condition is necessary and crucial for achieving the PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves both the positivity and the DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples further confirm the theoretical findings.

We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in infinite-dimensional space, maps the solution from a current time to a future time and completely characterizes the solution evolution of the underlying unknown PDE. Our recovery strategy relies on approximation of the evolution operator in a properly defined modal space, i.e., generalized Fourier space, in order to reduce the problem to finite dimensions. The finite dimensional approximation is then accomplished by training a deep neural network structure, which is based on residual network (ResNet), using the given data. Error analysis is provided to illustrate the predictive accuracy of the proposed method. A set of examples of different types of PDEs, including inviscid Burgers' equation that develops discontinuity in its solution, are presented to demonstrate the effectiveness of the proposed method.ing studies are relatively limited, as they mostly focus on learning certain types of PDE or identifying the exact terms in the PDE from a (large) dictionary of possible terms. The specific novelty of this paper is that the proposed method seeks to recover/approximate the evolution operator of the underlying unknown PDE and is applicable for general class of PDEs. The evolution operator completely characterizes the time evolution of the solution. Its recovery allows one to conduct prediction of the underlying PDE and is effectively equivalent to the recovery of the equation. This is an extension of the equation recovery work from [17], where the flow map of the underlying unknown dynamical system is the goal of recovery. Unlike the ODE systems considered in [17], PDE systems, which is the focus of this paper, are of infinite dimension. In order to cope with infinite dimension, our method first reduces the problem into finite dimensions by utilizing a properly chosen modal space, i.e., generalized Fourier space. The equation recovery task is then transformed into recovery of the generalized Fourier coefficients, which follow a finite dimensional dynamical system. The approximation of the finite dimensional evolution operator of the reduced system is then carried out by using deep neural network, particularly the residual network (ResNet) which has been shown to be particularly suitable for this task [17]. One of the advantages of the proposed method is that, by focusing on evolution operator, it eliminates the need for time derivatives data of the state variables. Time derivative data, often required by many existing methods, are difficult to acquire in practice and susceptible to (additional) errors when computed numerically. Moreover, the proposed method can cope with solution data that are more sparsely or unevenly distributed in time. Since the proposed framework is rather general, we present seve...

The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics. The paper develops high-order accurate physical-constraints-preserving (PCP) central discontinuous Galerkin (DG) methods for the one-and two-dimensional special relativistic hydrodynamic (RHD) equations with a general EOS. It is built on the theoretical analysis of the admissible states for the RHD and the PCP limiting procedure enforcing the admissibility of central DG solutions. The convexity, scaling and orthogonal invariance, and Lax-Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form, and then the high-order central DG methods with the PCP limiting procedure and strong stability preserving time discretization are proved to preserve the positivity of the density, pressure, and specific internal energy, and the bound of the fluid velocity, maintain the high-order accuracy, and be L 1 -stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density or pressure etc.

This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal magnetohydrodynamics (MHD) on general meshes. Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of numerical MHD schemes with a Harten-Lax-van Leer (HLL) type flux on polytopal meshes in any space dimension. The main challenges overcome here include establishing certain relation between the PP property and a discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property. In the 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under a condition accessible by a PP limiter. For the multidimensional conservative MHD system, the standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence error in the magnetic field. We construct provably PP high-order DG and finite volume schemes by proper discretization of the symmetrizable MHD system, with two divergence-controlling techniques: the locally divergence-free elements and an important penalty term. The former technique leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals in theory that a coupling of these two techniques is very important for positivity preservation, as they exactly contribute the discrete divergence terms which are absent in standard multidimensional DG schemes but crucial for ensuring the PP property. Several numerical tests further confirm the PP property and the effectiveness of the proposed PP schemes. Unlike the conservative MHD system, the exact smooth solutions of the symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint.