Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws

Imre Fekete; Ferenc Izsák; Vendel P. Kupás

math.NASep 26, 2026

Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws

Authors: Imre Fekete, Ferenc Izsák, Vendel P. Kupás

Organizations: Department of Applied Analysis and Computational Mathematics, ELTE Eötvös Loránd University, Budapest, Hungary · Department of Network and Data Science, Central European University, Vienna, Austria · Department of Applied Analysis and Computational Mathematics and NumNet HUN-REN–ELTE Research Group, ELTE Eötvös Loránd University, Budapest, Hungary

Abstract

A clear link is established between conventional numerical methods and neural network approximations for solving one-dimensional scalar conservation laws. The focus is on the construction of an appropriate flux term in the case of convex flux functions for improving the classical schemes. The first neural network developed here is able to rediscover Godunov's method, while the second one emulates the behavior of a second-order slope-limiter function. In this way, by merging them, second-order reconstruction-based schemes can be developed. The networks presented here employ a minimal number of parameters, significantly reducing the complexity compared to previous approaches. These networks can also be linked consecutively to get a deep one corresponding to multiple time steps. Training them with an appropriate loss leads to stable schemes, improving even the classical methods without increasing their complexity.

Figures & tables

Panel of Figure 1 — Figure 1: The piecewise linear, continuous function $\Phi$ with breakpoints $x_{0}=0<x_{1}<x_{2}<x_{3}<x_{4}=1$ . Function values $\Phi(x_{i})$ are shown at each node, with slopes $m_{k}$ on each interval.

Panel of Figure 2 — Figure 2: The residual block for computing a single time step using a skip connection.

Panel of Figure 3 — Figure 3: The neural network for the fully discretized numerical solution corresponding to ( 2 ).

Panel of Figure 4 — Figure 4: Partition of the first quadrant into regions $T_{0}$ to $T_{3}$ based on linear inequalities for $a$ and $b$ in ( 9 ).

Panel of Figure 5 — Figure 5: A diagram showing the subnetwork, which generates the reconstructed slopes with $s=s_{1}+s_{3}$ .

Figure 6: Learning the Godunov scheme. Left: the Frobenius norm of the difference of the matrix

A

in Proposition 2 and the one in the training process. Right: training and validation losses over 100 epochs.

Figure 7: Result of the full simulation corresponding to Figure 2 using the initial matrix

A_{0}

and the final one after 100 epochs. The trigonometric test problem was used, with the simulation run for 0.1875 time units after the shock developed.

			Number/size of
Problem	Method	Computer	Input $\&$ output	Layers	Para- meters	Epochs	Time
Burgers	Godunov	Laptop	128	64	4	100	355 s
Burgers	Godunov	HPC	128	64	4	100	235 s
LWR	Godunov	Laptop	128	64	4	100	477 s
LWR	Godunov	HPC	128	64	4	100	315 s
Burgers	reconstruction	Laptop	128	64	5	20	240 s

Table 1: Summary of the main characteristics of neural networks and performance of the learning process for getting optimal conservative schemes in case of the Burgers and the LWR equation.

Panel of Figure 8 — Figure 8: Variation of slope limiters in the training procedure using two different initial parameters over 20 epochs. Left: starting from the superbee limiter, right: starting from the minmax limiter. In both cases, we used the parameters nx = 129 and $\delta=\frac{0.25}{64}$ .

Panel of Figure 9 — Figure 9: Learned slope limiters obtained with fixed spatial discretization nx = 257 (left) and with fixed CFL number 0.25 (right).

Panel of Figure 10 — Figure 10: Learned slope limiters obtained for trigonometric initial conditions (left) and fixed final time $dt=0.5$ (right).

CFL	Limiter	$nx$
CFL	Limiter	33	65	129	257
0.5	Minmod	$9.20$	$5.90$	$4.01$	$2.71$
	MC	$9.35$	$7.33$	$5.91$	$4.82$
	Superbee	$12.35$	$10.35$	$8.17$	$6.40$
	Learned	7.68	4.81	3.19	2.18
0.25	Minmod	$10.23$	$6.08$	$3.73$	$2.26$

Table 2: Discrete

L_{2}

errors for different limiters and CFL numbers. The reported values are scaled by

10^{-3}

Initial condition	Limiter	$\Delta t$
Initial condition	Limiter	0.25	0.5	1.0
Riemann	Minmod	$5.51$	$5.07$	$4.65$
	MC	$4.72$	$5.31$	$6.31$
	Superbee	$5.90$	$7.53$	$9.65$
	Learned	4.65	4.53	4.31
Linear	Minmod	$2.93$	$3.55$	$3.48$

Table 3: Experimental comparison of different slope limiters: discrete

L_{2}

errors for various initial conditions and final times. Entries are given in units of

10^{-3}

Figure 11: Comparison of the neural network-based limiter with the exact solution, the Godunov scheme, and classical flux limiters for a Riemann problem.

CFL	Method	$nx$
CFL	Method	101	201	401
0.5	CNN_FLOW	$17.40$	$11.81$	$8.22$
	NN_R	8.06	$5.87$	$4.60$
	NN	$8.13$	5.86	4.52
0.25	CNN_FLOW	$20.09$	$13.71$	$9.52$
	NN_R	$7.72$	$4.81$	$3.15$

Table 4: Discrete

L_{2}

errors obtained by different neural network-based methods. Entries are given in units of

10^{-3}

Ref.	Basic approach	Number/size of
		layers	neurons per layer	parameters	Remark
[ 26 ]	PINN	3	20	$\sim$ 1,300	also: comparison
[ 23 ]	PINN	4	different	$1200$	foundational work
[ 6 ]	PINN	5	50	$10401$	based on weak form
[ 15 ]	FNO	3	40	$3441$	foundational work
[ 3 ]	FNO	5	20	$1761$	special treatment for discontinuities

Table 5: A summary of some neural network methods, architectures, and computational complexity for solving the one-dimensional Burgers equation.

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A second order finite volume scheme rests on two local quantities: a gradient reconstructed in each cell, and a limiter which scales it down where the reconstruction would overshoot. Both are set by fixed formulas, and on coarse unstructured meshes a small network can supply better values. But a network is free to output anything, and the usual safeguard is a penalty in the training loss, which discourages inadmissible states without preventing them. We replace the penalty by a hard constraint. The network still sets both quantities, and every value it can produce lies inside safe bounds: its stencil weights cannot cancel a neighbour, and its limiter is capped by the local flow. The flux, the wall treatment and the time step are not learned and carry their own guarantees. Admissibility therefore holds for every value of the weights rather than as an outcome of training, and no negative density or pressure occurred in any computation reported here. Because the scheme is safe whatever the network does, we could ask what the network contributes. We test it on supersonic channel flow over an obstacle, including the forward facing step of Woodward and Colella. Learning lowers the error by 38% on an unseen geometry and 29% on an unseen obstacle topology, measured against the same scheme with the network switched off. The method aims at the accuracy of a fine mesh for the cost of a coarse one, and refining once improves the error fourfold while multiplying the run time by eight. Learning secures half of this improvement for a sixth of this time. All of this comes from one of the two quantities the network sets. The gradient reconstruction reproduces the full effect on its own, and the limiter accounts for about a tenth as much. This also explains why the gain fades beyond the Mach numbers the weights were trained on.

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