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Towards Exact Multidimensional Fluxes: A Structure-Preserving Data-Informed Framework for Riemann Solvers

Yucheng Zhang, Clint Dawson

arXiv:2610.35265v1physics.comp-phcs.NAmath.NA

TL;DR

Exact multidimensional Riemann fluxes are difficult to evaluate, motivating a solver that learns polynomial flux coefficients from initial states while enforcing structural properties. In two-dimensional Euler tests, it remains stable at CFL 0.9 where the exact one-dimensional solver fails, and compares favorably in accuracy and computational time.

  • Problem

    In multidimensional Riemann problems, complex wave interactions make the exact space-time flux integral difficult to evaluate, while face-normal solvers omit transverse coupling.

    Source: §1 · p. 2

  • Method

    The framework learns coefficients of a polynomial representation of the flux integral from initial states using a neural network trained on offline high-resolution simulations, while enforcing consistency and symmetry properties.

    Source: §1 · p. 3, §5 · p. 21

  • Results

    At CFL 0.9, MH+SPIDERS remains stable while MH+ExactRS fails; against MH+ExactRS at CFL 0.45, its L1 errors are 8–15% lower and it takes about 30% less time on finer grids.

    Source: §5 · pp. 21–22

  • Takeaways & Limitations

    In the tested two-dimensional Euler problems, SPIDERS is a more accurate, faster alternative to MH+ExactRS while preserving radial symmetry in the cylindrical explosion.

    Source: §5 · pp. 21–22

    Accuracy outside the training distribution is not guaranteed; the framework was demonstrated only for two-dimensional Euler and compared only with the exact one-dimensional Riemann solver.

    Source: §5 · p. 22

Abstract

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Finite-volume and discontinuous Galerkin methods update cells using the space-time flux integral over their faces, which typically depends on the solutions to local Riemann problems. In the multidimensional Riemann problem setting, this integral is difficult to evaluate exactly because of complex wave interactions. In this work, we show that the exact space-time flux integral is a polynomial in the time step and face dimensions. This motivates a learning-based approach, as only the mapping from the initial states to the polynomial coefficients needs to be learned. We propose a structure-preserving, data-informed Riemann solver framework that uses a neural network trained on high-resolution simulation data to learn this mapping. Key structural properties, including consistency and symmetry, are enforced by construction. Provided that the high-resolution simulations converge to the exact flux integrals and the network accurately predicts the coefficients, this framework yields near-exact flux integrals for use in numerical simulations. We demonstrate the framework using classical implosion and explosion problems for the two-dimensional Euler equations. In an unsplit MUSCL-Hancock scheme, the learned Riemann solver remains stable at a CFL number of 0.9, whereas the same scheme using the exact one-dimensional Riemann solver fails. Compared with the latter scheme at CFL=0.45, the learned scheme is slightly more accurate and requires approximately 30% less computational time.

1. Introduction

Multidimensional face-normal solvers omit transverse wave interactions, while exact nonlinear multidimensional fluxes are difficult to obtain. The paper proposes learning polynomial flux coefficients from initial states to address this challenge.

1. Introduction

  • Without transverse coupling, face-normal Riemann solvers miss transverse wave interactions, contributing to mesh imprinting and a stable time step that decreases with spatial dimension.Source: §1 · p. 2
  • For nonlinear systems such as Euler, the two-dimensional Riemann problem lacks a closed-form solution, and high-resolution solves at every vertex and time step are too costly.Source: §1 · p. 2
  • The learning target is challenging because the space-time flux integral depends on initial states, time step, and face size.Prior work introduced HCNRS, which enforces consistency, symmetry, and scaling invariance for Riemann solvers [21].Source: §1 · p. 3
  • The paper shows that, when waves stay within the face during the time step, the exact flux integral is polynomial in time step and face size, reducing learning to coefficients from initial states.The stated condition holds for CFL numbers up to one; the proposed Euler solver enforces consistency, mirror symmetry, scaling invariance, and rotational invariance.Source: §1 · p. 3

2. Polynomial dependence of the space–time flux integral on the time step and face dimensions

Under self-similarity, finite domain of dependence, and integrable flux assumptions, the space–time face flux has a polynomial structure when tangential Courant numbers are at most one. The result extends from one-dimensional fluxes to face corrections in two and three dimensions and an inclusion–exclusion formula in arbitrary dimensions.

2. Polynomial dependence of the space–time flux integral on the time step and face dimensions

  • The derivation assumes self-similarity, a finite domain of dependence, and well-defined, integrable normal fluxes; on Cartesian faces, each tangential Courant number must be at most one.The result is derived dimension by dimension and extended using inclusion–exclusion.Source: §2 · pp. 3–4

2.1. One space dimension

  • In one dimension, the space–time flux integral is a degree-one polynomial in ∆t, with no spatial degree because the face is a point.The flux at the interface is time-independent by self-similarity.Source: §2.1 · p. 4

2.2. Two space dimensions

  • In two dimensions, the face flux is the one-dimensional flux plus a vertex correction proportional to t, and the integrated correction depends only on the four initial states.The stated condition is λ∆t ≤ ∆y.Source: §2.2 · pp. 4–5

2.3. Three space dimensions

  • In three dimensions, the face flux decomposes into one-dimensional, two edge, and vertex contributions, with the vertex correction scaling as t^2.The restrictions require λ∆t ≤ min(∆y, ∆z).Source: §2.3 · p. 8

2.4. Any number of space dimensions

  • In d dimensions, the space–time flux integral has degree at most d in ∆t and at most one in each tangential face length; every nonzero monomial has total degree d.This polynomial structure holds within the stated time-step restrictions.Source: §2.4 · p. 10

3. Neural network architecture

The solver targets the integrated flux difference for the two-dimensional Euler equations and builds structural constraints into its network. Rotation, scaling, consistency, and mirror symmetry are handled through transformations and input reduction.

3. Neural network architecture

  • The two-dimensional Euler solver enforces consistency, mirror symmetry, scaling invariance, and rotational invariance by construction.Consistency reduces one-dimensional-across-the-face data to the one-dimensional Riemann flux.Source: §3 · pp. 10–11

3.1. Rotational invariance

  • The same network predicts the integrated flux difference for each face by rotating the vertex problem so that the target face points up, then rotating the output back.The other three face results are obtained by rotation.Source: §3.1 · p. 11

3.2. Scaling invariance

  • An invariant-based reduction represents the solver inputs using 14 variables.The normalized density and pressure dependencies are represented by three dimensionless contrasts each.Source: §3.2 · p. 12

3.3. Consistency and mirror symmetry

  • Consistency is imposed by multiplying state jumps across the face by a network output, making the correction vanish for one-dimensional data; mirror symmetry is enforced jointly by symmetrization.The mirrored problem exchanges left and right quadrants and changes the signs of selected flux components.Source: §3.3 · p. 13

4. Numerical results

The numerical results evaluate SPIDERS’ training data, model, and use in an unsplit MUSCL–Hancock scheme. Across implosion and explosion tests, the scheme is compared with MH+ExactRS for accuracy, stability, and computational cost.

4.1.1. Training Data Generation

  • Training data combine independently drawn states for global coverage with perturbed, similar states; 540,000 problems with close states and 60,000 with independent states yield four face samples each.State ranges reflect the target simulations, and the dataset excludes problems whose solutions would contain vacuum.Source: §4.1.1 · pp. 14–15

4.1.2. Neural Solver Model

  • SPIDERS uses a four-hidden-layer MLP and achieves validation mean relative errors for D of 0.7% on close-state problems and 2.5% on independent-state problems.Training uses Adam, a cosine learning-rate schedule, and the network with the lowest validation loss.Source: §4.1.2 · p. 15

4.2. Numerical scheme

  • MH+SPIDERS combines the exact one-dimensional face flux with two vertex corrections computed from the four evolved corner states.The schemes share the MUSCL–Hancock predictor and differ only in the face flux.Source: §4.2 · pp. 15–16

4.3. Implosion

  • The implosion schemes produce similar reflected shocks and contact discontinuities, while the jet head reaches about 0.158 with MH+SPIDERS versus 0.149 with MH+ExactRS.The comparison uses 400 × 400 cells; MH+SPIDERS runs at CFL 0.9 and MH+ExactRS at CFL 0.45.Source: §4.3 · p. 16, Figure 3 · p. 17

4.4. Cylindrical explosion

  • In the cylindrical explosion, MH+SPIDERS has lower L1 error than MH+ExactRS for density, radial velocity, and pressure, while both schemes show grid imprinting at the contact.Across the domain, the errors are lower by 9.0%, 14.9%, and 13.5%, respectively; Figure 5 compares profiles and cumulative error against the radial reference.Source: §4.4 · pp. 17–18, Figure 4 · p. 18, Figure 5 · p. 19

4.5. CFL number and computational cost

  • At full GPU utilization, MH+SPIDERS uses about 30% less time; MH+ExactRS fails at CFL 0.75 in implosion and 0.8 in explosion.Figure 6 plots density error against CFL, and Table 2 reports wall times for both schemes.Source: §4.5 · pp. 18, 20, Figure 6 · p. 20, Table 2 · p. 21

5. Conclusions

The exact multidimensional space–time flux integral is polynomial in the time step and face dimensions for CFL numbers up to one, reducing the learning target to coefficients determined by initial states. The framework’s two-dimensional Euler tests show stability at CFL 0.9 and improved accuracy and runtime against the same scheme using the exact one-dimensional Riemann solver, subject to limits on training coverage and evaluation scope.

5. Conclusions

  • For any number of space dimensions and CFL numbers up to one, the exact space–time flux integral is polynomial in the time step and face dimensions.Learning therefore reduces to mapping initial states to polynomial coefficients.Source: §5 · p. 21
  • The neural framework enforces consistency, mirror symmetry, and rotational invariance, and approaches the exact flux integral if simulations converge to it and coefficient predictions are accurate.Source: §5 · p. 21
  • MH+SPIDERS is stable at CFL 0.9, where the same MUSCL–Hancock scheme with the exact one-dimensional Riemann solver fails, and is more accurate while using half as many time steps.In the cylindrical explosion, L1 errors are 8–15% lower on grids from 101 × 101 to 801 × 801 cells; on finer grids, it takes about 30% less computational time.Source: §5 · pp. 21–22
  • SPIDERS accuracy is not guaranteed for states outside its training distribution; the framework was demonstrated only for two-dimensional Euler equations and compared only with the exact one-dimensional Riemann solver.The authors identify three-dimensional and more complex systems, near-vacuum states, and comparisons with multidimensional solvers as future work.Source: §5 · p. 22

CRediT Authorship Contribution Statement

Yucheng Zhang’s listed contributions span the technical work and writing, while Clint Dawson’s include supervision, project administration, and funding acquisition.

CRediT Authorship Contribution Statement

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