A diffusion-free multi-layer neural-network method for multidimensional nonlinear hyperbolic equations

1. Introduction

• Neural networks are used for parameterized PDE solvers, with PINNs among popular approaches and external experimental or numerical data available to the solver.The introduction also cites alternative methods and identifies as pioneering work generalized by PINNs.Source: §1 · p. 1
• Shocks are difficult for standard methods: finite-volume schemes can introduce artificial diffusion, while direct smooth-network approximations produce regularized shocks.Higher-order finite-volume approximations can reduce diffusion but raise stability issues requiring slope limiters.Source: §1 · pp. 1–2
• PINN-like methods can approximate smooth HCL solutions, including rarefaction waves, but stability issues may shift to optimization and integration time must remain limited to avoid a null solution.The methods reformulate initial-value problems as boundary-value problems, treating the initial condition as a boundary condition.Source: §1 · p. 2
• LeafNet approximates smooth solutions and Rankine–Hugoniot jumps, then reconstructs piecewise-smooth shock solutions, extending the one-dimensional algorithm [18].Its stated scope includes N-dimensional conservation-law systems and self-similar rarefaction networks; wave interactions, overturning, and closed surfaces are deferred.Source: §1 · p. 2

2. Scalar conservation laws in N dimensions

• The problem is an N-dimensional nonlinear scalar conservation law posed on Q = Ω × [0, T], with N ≥ 1.The domain Ω is an open smooth subset of R^N and may be bounded or unbounded.Source: §2 · p. 3
• The scalar state u takes values in R, and the flux f(u) is the derivative F′(u) of a C2 potential F.The flux has N components, one for each spatial dimension.Source: §2 · p. 3
• Boundary conditions are imposed where characteristic lines are incoming; the whole-space setting can use a compactly supported smooth extension of the initial data and a reduced time interval.The text also considers rectangular domains.Source: §2 · p. 3

2.1. Shock waves and the principle of the LeafNet method

• LeafNet tracks the shock as a graph g, solves an (N−1)-dimensional Rankine–Hugoniot IVP alongside independent smooth-side IBVPs, and reconstructs the full solution piecewise.The construction extends the shock-wave method presented in for N = 1, 2 to dimension N and provides C1 estimates for u±.Source: §2.1 · p. 5

2.1.1. Neural network algorithm: the LeafNet algorithm

• The LeafNet algorithm combines PINNs to approximate the smooth side solutions u± and shock graph g, avoiding spurious regularization of discontinuous waves.The approximate full solution selects u− or u+ according to whether xN lies below or above g.Source: §2.1.1 · pp. 6–7
• The algorithm does not hard-constrain entropic waves; entropy conditions could be added to the loss, but would deteriorate neural-network training.Entropy questions were also briefly addressed in.Source: §2.1.1 · p. 7

2.1.2. Theoretical justification of Algorithm 2.1.1

• For sufficiently small T under Theorem 2.1's assumptions, the side IBVPs have unique C1 solutions with bounded first derivatives, and the Rankine–Hugoniot IVP yields a unique piecewise-C1 solution with a C1 shock surface.The theorem extends the existence result to dimension N and supplies L∞ estimates used to assess approximate shock waves.Source: §2.1.2 · p. 7
• Theorem 2.2 establishes local unique classical solutions for the stated quasilinear first-order PDEs, with L∞ bounds on the solution and its first derivatives.These estimates support the proof of the LeafNet method's existence and regularity results.Source: §2.1.2 · p. 8
• For sufficiently small T, Γ is a C1 surface that is the shock wave associated with the conservation law.The proof uses the characteristic condition to show that Γ intersects only pairs of characteristics, one from each side of the initial interface, and is independent of the extensions u±.Source: §2.1.2 · p. 11

2.1.3. LeafNet neural network approximation error estimate for scalar shock waves

• The shock-location error satisfies a first-order PDE to which the estimate applies, even though its right-hand side is only C0.The flux remains C1 and satisfies the required assumptions.Source: §2.1.3 · p. 13
• The L∞ error in approximating the shock is bounded by Sobolev norms of the network residuals σi, σb, and ε±.The bound is stated in terms of problem data and these residuals, and also applies on suitable bounded domains.Source: §2.1.3 · p. 15

2.2. Shock wave formation

• The setup assumes no initial shock and exactly one shock forming at t* and developing through T.The corresponding discontinuity surface is denoted Γ.Source: §2.2 · p. 15
• At a time t1 after formation, the shock cross-section Γ1 is assumed to be the graph of a C1 function g1.If u1 were known, the tracking procedure could restart from u1 and Γ1.Source: §2.2 · p. 15
• Because u1 and Γ1 are unknown, a virtual discontinuity Γ0 initializes u±; the method then solves for u± and g.Before t*, g can solve the interface equation; after t*, it must satisfy Rankine–Hugoniot and parameterizes Γ.Source: §2.2 · pp. 15–16

2.3. Neural network solution for rarefaction waves

• Classical PINNs provide a smooth approximation that is inaccurate near the initial discontinuity developing into a rarefaction wave.The method seeks a sharp approximation even near that discontinuity.Source: §2.3 · p. 16

2.3.1. Discontinuous rarefaction waves

• The rarefaction region Qrw lies between characteristic surfaces Γ− and Γ+, and a normalized coordinate sN maps it to (0, 1).The transformation sends xN between g− and g+ to sN; the inverse maps back to xN.Source: §2.3.1 · p. 16
• In normalized coordinates, w satisfies h_i∂x_i w + h_N∂s_N w = 0 on Ω′×(0,1)×(0,T), with boundary values inherited from u− at s_N=0 and u+ at s_N=1.Source: §2.3.1 · p. 17

2.3.2. Neural network algorithm for rarefaction waves

• The neural implementation uses networks u± for the branch solutions, g± for the curved faces, and w for the transformed rarefaction solution.When the faces cannot be solved analytically, the algorithm seeks network approximations; it then solves for w and reconstructs u.Source: §2.3.2 · p. 18, §2.3.1 · p. 18

2.4. Systems of hyperbolic conservation laws in N dimensions

• The system formulation considers vector-valued conservation laws with incoming-flux boundary conditions and initial data in L1 and TV.The system unknown has m components and the flux maps into Rm×N.Source: §2.4 · pp. 19–20

2.4.1. Shock waves for the systems of conservation laws

• For systems, the Rankine–Hugoniot condition is imposed componentwise, yielding equations for the shock graph g.The algorithm minimizes the summed losses of the equations to construct g, then reconstructs the solution from independent u− and u+ solves; networks are vector-valued.Source: §2.4.1 · p. 20

2.4.2. Shock wave formation

• Shock formation tracking starts from a virtual discontinuity, approximates u± and the Rankine–Hugoniot solution g, and identifies t* as the earliest discontinuity in the jump.Source: §2.4.2 · p. 21

3. Numerical experiments

• The experiments use fully connected tanh networks, H1-norm loss functions, gradient descent, and initial and boundary conditions encoded directly in the networks.Most computations use a learning rate of 2 × 10^-3.Source: §3 · p. 21

3.1. Scalar HCL

• LeafNet represents scalar shock discontinuities sharply without artificial numerical diffusion and supports independently computed local solutions suited to parallel architectures.The experiments report accurate shock approximation by exploiting the solution’s mathematical structure.Source: §3.1 · pp. 22, 25
• Self-similar networks accurately approximate a one-dimensional rarefaction wave, while direct PINN solutions artificially regularize its non-smooth regions.The comparison uses the same number of layers and neurons in both methods.Source: §3.1 · p. 24
• For a two-dimensional rarefaction wave, self-similar networks provide an accurate approximation without artificially regularizing non-smooth regions.The experiment reports the solution at T = 1.1.Source: §3.1 · p. 25
• LeafNet accurately generates and propagates a shock formed from smooth initial data in the two-dimensional scalar experiment.The reported shock-generation time is t* = 0.2.Source: §3.1 · p. 27

3.2. Systems of HCL

• In a two-dimensional shallow-water system, the method propagates a single shock without artificial diffusion.The experiment is a preliminary illustration focused on one shock, rather than arbitrary or more complex initial wave decompositions.Source: §3.2 · pp. 27, 30
• A further shallow-water experiment tests LeafNet with a non-straight initial discontinuity.It changes the initial interface to g0(x2) = β(x2 − a2), with β = 10^-1.Source: §3.2 · p. 30

3.3. About the entropy of the LeafNet method

• LeafNet does not explicitly impose an entropy constraint, and its direct single-shock application does not capture the exact compound-wave solution for the tested non-convex flux.The example’s entropy solution consists of a shock followed by a rarefaction, with an intermediate state.Source: §3.3 · pp. 31–33

• The method addresses piecewise-smooth solutions of N-dimensional conservation laws, including shock tracking, shock formation, and rarefaction waves.Source: §4 · p. 33
• For shock waves, the method is shown to be well-posed, with an error estimate for the shock-wave approximation.Source: §4 · p. 34
• Reformulating the laws as a coupled system of smooth local solutions provides a setting compatible with neural networks approximating PDEs in strong form.Source: §4 · p. 34