(U)NFV: (Un)Supervised Neural Finite Volume Methods for Solving Hyperbolic PDEs
Authors: Nathan Lichtlé
Alexi Canesse
Zhe Fu
HOSSEIN NICK ZINAT MATIN
Maria Laura Delle Monache
Alexandre M Bayen
Venue: ICLR 2026 Poster (2026)
Keywords: Neural PDE solvers
Hyperbolic conservation laws
Finite volume methods
Physics-informed learning
Traffic modeling
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
TL;DR: We propose (U)NFV, a neural finite volume framework that learns conservation-law dynamics, achieving higher accuracy and scalability than classical PDE solvers for shocks, discontinuities, and traffic modeling.

Abstract: We introduce (U)NFV, a modular neural network architecture that generalizes classical finite volume (FV) methods for solving hyperbolic conservation laws. Hyperbolic partial differential equations (PDEs) are challenging to solve, particularly conservation laws whose physically relevant solutions contain shocks and discontinuities. FV methods are widely used for their mathematical properties: convergence to entropy solutions, flow conservation, or total variation diminishing, but often lack accuracy and flexibility in complex settings. Neural Finite Volume addresses these limitations by learning update rules over extended spatial and temporal stencils while preserving conservation structure. It supports both supervised training on solution data (NFV) and unsupervised training via weak-form residual loss (UNFV). Applied to first-order conservation laws, (U)NFV achieves up to 10x lower error than Godunov's method, outperforms ENO/WENO, and rivals discontinuous Galerkin solvers with lower implementation burden. On traffic modeling problems, both from PDEs and from experimental highway data, (U)NFV captures nonlinear wave dynamics with significantly higher fidelity and scalability than traditional FV approaches.

Published as a conference paper at ICLR 2026
(U)NFV: (UN)SUPERVISED NEURAL FINITE VOLUME METHODS FOR SOLVING HYPERBOLIC PDES
Nathan Lichtlé',,, Alexi Canesse 1,,,, Zhe Fu Hossein Nick Zinat Matin "" Maria Laura Delle Monache', Alexandre M. Bayen'
University California, Berkeley, "école polytechnique, LIX, IP Paris, CNRS [lichtle, alexi.canesse, zhefu, h-matin, midellemonache, bayen @ @errkcley.ddu
ABSTRACT
We introduce (U)NFV, a modular neural network architecture that general- izes classical finite volume (FV) methods for solving hyperbolic conservation laws. Hyperbolic partial differential equations (PDEs) are challenging to solve, particularly conservation laws whose physically relevant solutions contain shocks and discontinuities. FV methods are widely used for their mathematical properties: convergence to entropy solutions, flow conservation, or total variation diminishing, but often lack accuracy and flexibility in complex settings. Neural Finile Volume addresses these limitations by learning update rules over extended spatial and tem- poral stencils while preserving conservation structure. It supports both supervised training on solution data (NFV) and unsupervised training via weak-form residual loss (UNFV). Applied to first-order conservation laws, (U)NFV achieves up 10x lower error than Godunov's method, outperforms ENO/WENO, and rivals discontinuous Galerkin solvers with lower implementation burden. On traffic mod- eling problems, both from PDEs and from experimental highway data, (U)NFV captures nonlinear wave dynamics with significantly higher fidelity and scalability than traditional FV approaches. Code, dataset, trained models, and videos can be found http:s//maaaanliccccc..o///eeee
u(t = 1,+)
Exact Solution
Godunov
NFV (Ours)
0, xz)
Figure Prediction of entropy solutions hyperbolic PDEs. Top: NFVå prediction vs. Godunov scheme for Burgers' equation at a fixed time. LWR predictions follow similar trend. Mid: Entropic solution u(t, x)
for Burgers' equation over domain (t, x) [0, 1Ja. Bottom: Corresponding initial condition u(0,-) +).
INTRODUCTION
Hyperbolic partial differential equations (PDEs) are fundamental tools to model propagation and transport phenomena with nonlinear or discontinuous behavior, appearing in areas like fluid dynamics and traffic flow. this work, we focus on an essential subclass: conservation laws, which encode the
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principle that certain physical quantities, such as mass, momentum, energy, must be preserved over time. A general one-dimensional scalar conservation law takes the form:
0 0, t) + 0zf (u(z, t)) O,
(1)
where u is the conserved quantity and f is the flux function.
The solutions of hyperbolic PDEs are difficult to approximate due to discontinuities such as shocks, even when starting from smooth initial conditions (Evans) 2022). Consequently, classical (smooth) solutions typically cease exist after finite time, and must instead rely weak solutions. Closed-form solutions exist only in rare cases, such as on simple Riemann Levque (2002) through the Lax-Hopf formula (Lax 1957 Claudel and Bayen| 2010a b) in specific concave convex settings, As result, most practical applications rely on numerical methods for approximating the PDE solution, with finite volume (FV) methods (LeVeque) 2002) being a popular choice due to their ability to track conserved quantities across discontinuilies and capture shock dynamics.
Classical FV methods involve important trade-offs between accuracy near discontinuities, compu- tational cost, stencil size, and implementation complexity. In recent years, neural networks have been explored as flexible and powerful alternatives solvers, showing promise learning complex dynamics from data residuals. Yet, many such methods are designed for non-specific models, often the expense of losing physical structure, including conservation laws and entropy behaviors.
We introduce the Neural Finite Volume (NFV) method, modular architecture tailored to conservation laws, that blends the prreeernn- benefits of FV schemes with the expressiveness of neural networks. Conservation is built into the NFV model, using extended spatial and temporal stencils. We develop both a supervised version, trained on solution data from simple cases, and an unsupervised variant (UNFV), which learns directly from the PDE via a weak-form residual loss. This flexibility allows (U)NFV to adapt to the availability of data, leveraging accurate synthetic or field data when present, solving directly from the equation when solutions We focus on one-dimensional scalar conservation laws, which are widely used in applications such as traffic flow, pipeline and channel models, and form standard, well-understood testbed.
inaccurate or expensive to obtain.
Contributions. Our main contributions are as follows:
We propose (U)NFV, neural architecture that generalizes the structure of finite volume methods and thus preserves conservation properties by construction.
We introduce two variants: a supervised learning one (NFV) and an unsupervised learning one (UNFV), depending data availability, using either solution data or a weak-form residual loss. We demonstrate strong numerical results on several conservation laws, achieving up to 10x lower error than classical FV solvers, shown in Figure !
Additionally, (U)NFV matches the accuracy of discontinuous Galerkin methods, without their mathematical complexity. We show that NFV be trained field data that does not strictly satisfy the conservation law, and still predicts accurate solutions with more generalizability than classical solvers.
The remainder of the article is organized follows: Section 2 provides detailed overview of the related work, Section 3 recalls the FV and introduces necessary notation, Section 4 describes the proposed (U)NFV method in detail, Section presentss the experiments and results on hyperbolic PDEs, Section 6 extends the NFV to experimental field highway data, and Section 7 concludes the article. Then, Appendix A provides details about FV schemes, Appendix B illustrates six PDE variants considered in this work, Appendix Cexxpnds on the experimental data handling and results from Section 6
and Appendix D details the model architecture, datasct, and hyperparameter choices.
2 RELATED WORK
Numerical methods. Classical numerical methods for hyperbolic PDEs, such as FV and discon- tinuous Galerkin (DG) (Hu and Shu 1999) methods, are widely used due to their capabilities in capturing shocks and discontinuities. First-order schemes such as the Lax-Friedrichs (Lax) 1954) and Godunov (Godunov) 1959b) methods provide robustness but suffer from excessive numerical diffusion. leading to smeared solutions. To address this, higher-orde methods like Essentially Non-O coan-siiiltttrry (ENO) (Shu 1999). Weighted ENO (WENO) (Liu et al. 1994), and DG have been introduced, offering improved accuracy in smooth regions while preserving stability near shocks. DG further improves accuracy through local polynomial approximations but incurs high computational
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costs (Cockburn and Shu 1998). In practice, DG and higher-order FV schemes like WENO demand intricate flux constructions, quadrature rules, and stabilization choices, whereas (U)NFV retains FV-like implementation complexity. Despite their accuracy, these methods often require extensive manual tuning effort, motivating the development of flexible, data-driven alternatives.
NN approaches for PDEs. Deep learning is powerful approach for approximating PDE solutions. In the supervised learning case, neural operators such as Fourier Neural Operator (FNO) (Li et al. 2020 and Deep Operator Networks (DeepONet) (Lu al. 2021) efficiently approximate solution mappings from parametric inputs, without requiring explicit mesh discretization in the case of FNO. However, these operators have mainly been validated on elliptic or parabolic PDEs, typically characterized by smooth solutions. Conventional neural architectures, such as CNNs (LeCun et al.) 1995) for structured domains and GNNs (Bronstein et al., 2017) for irregular geometries, have also been adopted, but supervised models rely heavily on large, high-quality labeled datasets, and often lack intrinsic enforcement physical constraints, leading to limited generalization and poor accuracy on PDEs involving sharp gradients shocks (Krishnapriyan al. 2021).
To reduce data reliance, unsupervised approaches like Physics-Informed Neural Networks (PINNs) incorporate PDE residuals directly into training losses (Raissi et al. 2017), proving effective for elliptic and parabolic equations (Raissi et al. 2019 Jagtap et al., al.] 2020)). However, PINNs encounter
significant difficulties with hyperbolic PDEs, especially in capturing discontinuities and shock dynamics, resulting in unstable optimization, convergence failures, and inaccurate solutions (Wang and Liu| 2021 Fuks and Tchelepi. 2020). Recent variants, such Weak PINNs (wPINNs) (De Ryck et al.) 2024), Parareal PINNs (PPINNs) (Meng et al., 2020). and Extended PINNs (XPINNS) (Jagtap and Karniadakis] 2020), aim overcome these issues through weak formulations specialized training strategies. Nonetheless, these adaptations often introduce significant complexity and require extensive hyperparameter tuning. underscoring need for methods tailored to hyperbolic PDEs.
NNs for hyperbolic PDEs and conservation laws. Neural approaches tailored to hyperbolic PDEs have introduced innovations handle shocks. Weak PINNS (wPINNs) (De Ryck et al.) 2024 integrate weak-form residuals or integral constraints to mitigate issues with discontinuities. Others employ neural networks directly within classical FV schemes to learn improved flux reconstruc- tions (Kossaczká et al. | Tong et al., 2024). However, these enhancements typically reintroduce complexity, such as extensive manual parameterization or problem-specific adaptivity, diluting the key advantage of neural flexibility and generality.
Motivated by these limitations, our proposed NFV approach learns local update rules directly from data or PDE residuals. By preserving the fundamental conservation-lav structure of traditional FV methods while flexibly leveraging neural networks, NFV achieves significantly higher accuracy, robustness, and scalability with minimal manual intervention.
3 PREREQUISITES AND NOTATIONS: FINITE VOLUME METHODS
Standard FV methods, such as those presented in Le Veque 2002), solve the integral form of the conservation law on mesh of uniform cells 1/21/21 T 1/2,, length Ar. The average of u over cell I, at time fn ntt for n = 1, 1,... ... NN and time discretization At, and the numerical flux through the interface 2/41/2 over the time step, are given respectively by
1,... mmaxx with cell
Ax / u(tnn x) dx and 1
1
phi1
u
f(ult, dt.
(2)
A first-order method JF approximates the numerical flux as F F(u! u²+1). while higher-order methods leverage additional cell averages. Let us generalize this framework by including cell averages from previous time steps in order construct even better approximations. Let FV1 be the class of meth- ods that use a rectangular stencil of a neighboring spatial cells times 6 past time steps estimate nu- merical fluxes. Specifically, define the (a, b)-ttecill
5
Figure 2: Example steneil for FV2, taking in a
stencil of 2 time steps times space cells.
at interface i + 1/2 as 14/,//2/0 (a, b)
{uu k= 2 2... and let an FV method JF estimate
m b.....--1 11
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numerical fluxes as F = (a Classical first-order methods, such as Godunov, fall
under class FV); more details about their computation are provided in Appendix To our knowledge, the vast majority of FV methods in the literature use single time step (i.e., 1) and small number of spatial cells. Indeed, designing analytical schemes with larger temporal or spatial stencils becomes exponentially more complex. Finally, the update rule given by the exact relation
At A
2
F 1411/2
PT 1//)
which in practice is approximated using the numerical fluxes E leading to an approximation oi of u.. Note that the influx of one cell is the outflux of another, which ensures conservation.
4 OUR METHOD: NEURAL FINITE VOLUME (NFV)
Our method builds upon the FV framework by using neural networks to approximate the numerical flux. Specifically, define NFV generalization of FV where the numerical flux Eu predicted by a neural network N based on local a x spatiotemporal stencil:
1 is
1/ = NN
The prediction of the solution is then updated using the classical FV update rule (3) ensuring mass conservation. We explore NFV models ranging from NFV; (matching Godunov's stencil) to NFV 10- using 11 spatial cells and 11 past time steps configurations that would be exceedingly complex to design manually due to the high-dimensional stencil involved. This extension enables accurate learning even from noisy field data. In practice, we implement NFV as a CNN (LeCun et al., 1995), which allows efficient computation across stencils due to the vectorized nature of CNNs. Since (UNNFV retains the standard finite volume update, boundary conditions such Dirichlet, Neumann, or open boundaries can be imposed via ghost cells prescribed interface fluxes exactly as in classical FV schemes, without modifying the neural architecture.
In all experiments we instantiate NFV as a lightweight two-dimensional CNN applied locally on each cell interface: the first layer uses a kernel of size over the spatial dimension with b input channels (one per time slice), followed by five X convolutional layers with 15 channels and either ELU or ReLU activations depending on the flux family. This architecture yields 1105 + 16(ab + 1) trainable parameters for NFV* a' so even our largest models contain only few thousand parameters while retaining the exact FV update rule.
We propose two variants of NFV that share the same architecture but differ in their training objectives: the supervised NFV a. trained on reference solutions, and the unsupervised UNFV trained directly from the PDE via a weak-form residual loss. The supervised setting applies when solution data is available. while the unsupervised variant enables training when such data is absent, relying instead on the governing conservation laws. Moreover, supervised NFV can also be applied in cases where the PDE is unknown but observational data is accessible, allowing solvers to be deployed directly on field data with only basic physical constraints, such as mass conservation, imposed, and without extensive hyperparameter tuning (see Section()).
In our experiments, we therefore train one (UNNFY model per conservation law, and once trained the same network can be applied to many different initial conditions for that equation, so the one- time training cost is largely amortized and in practice remains very short. At inference time, no optimization is solved: each time step is advanced by single application of the finite volume update rule (3) with numerical fluxes 1/ given by a forward pass the neural network, so the overall cost of solving an equation scales linearly with the number of time steps.
4.1 SUPERVISED LEARNING
Supervised learning offers a straightforward framework for training models when reference solutions available. In this study, employ supervised learning not only to approximate the solution of known equations but also to predict field data with unknown governing equations. Although solutions to hyperbolic PDEs are typically defined in the La space, consider their restrictions
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compact subsets where the functions are bounded, thereby allowing treatment within the L2 space. Accordingly, the loss function is defined as the standard mean square error.
L3 E
I
á|12
skg ag
where u is the true solution, i is the predicted solution, and R. is distribution over initial conditions.
4.2 UNSUPERVISED LEARNING
Unsupervised learning for hyperbolic PDEs particularly challenging because their solutions often lack closed-form expressions and classical (strong) solutions may not exist. Instead, these equations are typically defined through weak formulations. Although weak solutions are not unique: multiple functions can satisfy the PDE, but only one corresponds to the physically relevant entropy solution, which enforces admissibility conditions across shocks and discontinuities.
The unsupervised loss function is defined to minimize the residuals of the weak formulation, in order to approximate the entropy solution. While imposing this loss does not guarantee convergence to the entropy solution, empirical results indicate that method successfully converges to the entropy solution across various equations and numerous trials. To enhance learning efficiency, we optimize the weak formulation independently each time step by minimizing the squared residuals. The collection of test functions & consists of 250 randomly sampled, compactly supported polynomials of degree 50 over the spatial domain. The unsupervised loss reads:
E PEO
Σ
Σ
Imax
(At) (At)-
0. " J, 4 f(0))
1111/2 11-11112
Bo-R
where ûd denotes the predicted solution at spatial index and time step and R. is distribution over initial conditions. Note that for the scalar conservation laws considered here, integration by parts removes spatial derivatives from the weak-form loss, and time derivatives are handled via finite differences in the FV update, so no explicit spatial derivatives of the primal variables are required during training.
5
EXPERIMENTS
Experiments have been designed to answer four main questions:
Is (U)NFV compelling alternative classical finite volume methods?
Does UNFV converge to an entropy solution despite being trained on the weak formulation?
How does (UNNFV compare to much more complicated finite element methods?
Can NFV perform well on field data that contains noise and may be conservative?
5.1 BASELINES
Selecting appropriate baselines for PDE solvers poses challenges due to the diversity in computational frameworks: methods vary by mesh dependency (mesh-free free versus mesh-based), solution generation (autoregressive versus single-pass), and generalizability (operator-based versus retrained per initial condition). Therefore, we adopt classical numerical schemes, the foundation of our NFV method, baselines, ensuring fair comparison. Given the fact that NFV is developed based traditional first-order FV methods, the present work provides compelling case for replacing standard FV solvers with the simpler yet effective NFV method whenever FV methods are typically employed. We consider all the numerical schemes introduced in Section as baselines: first-order FV methods (Godunov, Sax-Friedrichs, and Engquist-Osine higher-order ones (ENO. WENO), and DG. finite- element method that is well-known for superior accuracy but suffers from computational burden. More details can be found in Appendix
5.2 EQUATIONS
The Lighthill- Sitll-Whithmmm Whitham-Richards model (Lighthill and Whitham 1955 Richards 1956), known as LWR, is first-order hyperbolic conservation law used to model traffic flow. is expressed as Oep + Oz (pu(p))
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Table 1: Performance comparison between neural network models and classical numerical schemes. Results are computed over the evaluation set of 1000 piecewise constant initial conditions. For each method, we report mean and standard deviation in L2 norm (mean((u - ()²)).
1st order FV
Higher order FV
FEM
2
2NFV
GD
LF
EO
ENO
WENO
DG
s.shiedds
1.3e 445
1.54
1.36 3
4.5e
6.44
6.46
3.
Tri.
1.4c 1o21
9,6e"140"
2.3e
2.06
1.96
2.66 10
Tri. 2
2.4e 2169
3.1e%aa
3.8e"2ae
1.4e
3.8e 22.3
5.8e 2109 2109
5.8e 2te
4.1e 200
Trapez.
1.1e"st" sta"
1.6e 2
2.1e =
2.5e 2no?
2.1e snad
6.2e saw"
5.3e Ste
2.9e St
G.berg
1.4e 1o0
5.30 1.04 2.4
4.9071ar
1.1e
1.26
264
U.wood
3.8e hes
lac4
9.2e 9.26-1a.4
1.1e
9.8e 12c6
5.96
Burgers
8.5e
1.3e
1.94
2.6e 21.3
2.76
2.8e
1.0e
where p is the density of the traffic, f pu(p) is the flux function and v is the velocity. The flux function is typically modeled as a concave function of the density. Variations the underlying velocity function give rise to different traffic flow models. In this work. six different models have been considered: Greenshields' (Greenshields et al.] 1935). Triangular (Geroliminis and Daganzo 2008), Triangular skewed (Geroliminis and Daganzo| 2008), Trapezoidal (Geroliminis and Sun 2011), Greenberg (Greenberg 1959) and Underwood (Underwood] 1961). These models behave very differently and should be considered different equations, shown in Figure Formulations and illustrations of those six models are given in Appendix B
The inviscid Burgers' equation is a well-known hyperbolic conservation law used in various
domains such as fluid mechanics (Burgers) 1939), non-linear acoustics (Lombard et al. 2013). gas dynamics (Panayotounakos and Drikakis 1995), and traffic flow (Musha and Higuchi 1978). We
refer the reader to Cameron (2011) for thorough introduction. It is expressed as
1 O, =0 0,u + 2
(5)
This equation can be written in the classical form of a conservation law using the flux function f 1- 1/2 u². Exact solutions Riemann initial conditions are also known for this problem. Visualization of some solutions, including videos, are available on jur webpage and in Figure [
Our experiments focus on one-dimensional conservation laws in this work. Demonstrating that NFV can consistently outperform classical schemes in ID establishes a strong foundation before tackling more complex systems. NFV architecture is, in principle, extendable higher dimensions, since neural networks naturally scale to -iche-diimessionll inputs. Extending NFV to multi-dimension
FAJN
PAIN
20
ENO
WENO
DG
FA.N
80
LF
ENO
WENO
FA.N
FMNN
100
ENO
WENO
DG
NPV!
100
96
96
NFV
100
100
NPV
100
UNEV!
100
100
100
90
90
UNEV
100 100 100
47
UNEV
100
100
100
0
0
GD
100
59
70
70
0
GD
0
100
88
43
GO
100
0
D
0
LF
0
0
LI
0
0
0
LF
0
0
0
38
100
70
70
BO
0
9
100
43
42
80
100
D
ENO
92
ENO
53
57
100
57
ENO
100
100
WENO
WENO
100
58
WENO
100
100,
100
100, 100
100
100 100
100
100, 100
(a) Greenshields
(b) Triangular
(6) Underwood
Figure 3: Comparison of numerical schemes across flow functions. Each cell shows the proportion of the evaluation set on which the row scheme outperforms the column scheme. DG, the only FEM tested, is rarely beaten. NFV! and UNFV; outperform other first-order schemes and rival higher-order ones, making them strong choices depending on the equation.

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Exact Solution
Godunov
NFV3 (Ours)
Solndon utt -,))
-0.2
0.8
0.6
-0.4
Solusion
0.4
-0.6
-0.8
0
Space (X)
0
Space (X)
Figure 4: Comparison of the final density of the Burgers' equation (left) and LWR triangular equation (right) for NFVİ and the Godunov Scheme. The proposed method displays an excellent approximation of the exact solution, capturing sharp features such as discontinuities and points of non ann-ifferentiabiiity. It contains some minor oscillations the solution, which not present in the Godunov scheme. The latter, however, fails capture the discontinuitics and points of non-differentiabiliy.. offering a very smoothed solution.
Table 2: Evaluation of NFV3 using piecewise constant initial conditions. Error is reported in L2 norm. NFV achieves outstanding performance, gaining up to an order of magnitude improvement compared to Godunov and WENO. Its performance is close DG, while keeping the implementation simplicity of finite volume method and the computational complexity of NFV.
Godunov
WENO
NFV
NFV
DG
Burgers'
1.8e 1 2m--4
2.6e
210-3
8.3e
23.-4
2.2e
11.-4
1.0e
Greenshields
1.1e
11
6.9e
1.2e
1.6e
4.2e
32
Triangular
2.2c
3 910-0
2.0c 920
1.3e
18c
2.9e
120
2.7e
lie
will introduce additional challenges (e.g., numerical stability, computational complexity, and coupled variables), which we identify as important avenues for future work.
5.3 DATASETS
Training is performed using solutions derived from Riemann problems, which are initial value prob- lems characterized by piecewise constant initial conditions with a single discontinuity (see Figure S for examples). These problems are fundamental in the study of hyperbolic PDEs and serve as essential test cases for numerical methods. For the scenarios considered in this work, analytical solutions to Riemann problems are available, making supervised learning possible. Evaluation is performed on a more complicated set of several hundred complex initial conditions to assess the model's generalization capabilities. These conditions consist of piecewise constant functions with ten discontinuities, giving rise to entropy solutions with multiple interacting shocks and rarefactions.
Exact solutions for these test cases are computed using the Lax-Hopf algorithm Lax 1957 Claudel
and Bayen 2010a b) finer grid (see Appendix D).
For the LWR benchmarks we train NFV autoregressively on 2048 randomly sampled Riemann problems with single discontinuity, using discretization parameters At 5 10 Ar 10-2 100 spatial cells, and prediction horizons that are progressively increased from 10 to 250 steps under robust CFL ratio of 0.5. Evaluation uses several hundred more complex piecewise-constant initial conditions whose exact solutions are computed on a finer grid with At = 10-4, Ar = 10-3, 200 cells, and 1000 time steps via the Lax-Hopf algorithm. Unsupervised UNFV models minimize weak-form residual loss using 250 compactly supported polynomials of degree 50 as test functions over the spatial domain. Note that we use uniform space-time time grids for simplicity and fair comparison to FV baselines, but the (U)NFV update depends only on cell volumes and interface fluxes and is therefore compatible with non-uniform or adaptive discretizations,
5.4 RESULTS AND DISCUSSION
Table reports Lg error for NFV. UNFV and baseline methods across the seven benchmark equations. Our models consistently outperform all first-order FV methods, and surpass ENO/WENO schemes on about half of the equations. As expected, the higher-order DG method achieves signif-
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10 10-1.5
* 2FV
* UNFV
*
Godunov
T/2 error
10
x
1.5
10-4
AI
10-3
Figure 5: Convergence plots on Greenshields' flux. The Le error is computed against the exact solution on the evaluation set for different mesh discretizations. We report both eror average and standard deviation, log-log scale. The dashed vertical line illustrates the discretization which NFV; and UNFV) were trained; the models generalize to smaller discretizations. The ratio St/ 0.1 remains constant as the mesh is refined.
icantly lower errors. Table 2 shows that NFV;, while as simple to implement as standard NFV., achieves up to 10x better accuracy, approaching the performance of DG. In this sense, (UNNFV offers DG-level accuracy but with much smaller FV-like implementation complexity, substantially faster inference, and training that typically completes within fifteen minutes, while using memory comparable to Godunov and significantly lower than DG, since only the numerical flux is learned while the rest of the finite volume solver remains unchanged.
Figure shows the fraction of test cases each method wins. NFV, and UNFV)) consistently surpass first-order FV methods. Against ENO/WENO, performance varies: models outperform some equations, match on others, and underperform in few, highlighting the complexity of benchmarking across diverse problem settings. Still, the fact that NFV3 and UNFV] consistently do better than first-order methods is seen as sign that the approach appears to converge well. Specifically, NFV and UNFV? consistently produce errors bounded by those of Godunov, emphasizing their robustness.
Since all methods use autoregressive prediction, evaluating performance at the final time step provides a good proxy for cumulative error. Figure shows that the prediction of NFV closely aligns with the exact solution, with only minor oscillations observed. Notably, NFV; effectively captures sharp discontinuities with high accuracy without relying on smoothing techniques, which are commonly employed in traditional FV methods to mitigate numerical artifacts. Qualitatively, across the large set of complex test initial conditions that span weak to strong shocks and rarefaction patterns, NFV predictions remain visibly less diffusive than the others FV schemes, with sharp features dissipating more slowly while preserving stability.
Ablation on discretization size: Classical numerical schemes are known converge the grid is refined. Figure 5 shows that NFV, and UNFV) consistently achieve lower error than Godunov, a scheme proven converge, across discretizations, suggesting that (U)NFV also converges to the entropy solution; the approximately linear trend in the log-log plot further indicates a polynomial convergence rate.
Ablation on CFL ratio: To further assess stability under different time step choices, we vary the CFL ratio while keeping the spatial grid fixed on Greenshields' LWR and report the resulting errors in Table 3 Across this range of CFL values, NFV3 consistently attains lower mean error and substantially reduced variance compared first-order FV baselines, and remains competitive with higher-order ENO and WENO schemes. The only exception is DG at very small CFL, which achieves the lowest error but becomes unstable and fails to run higher CFL ratios, whereas NFV. remains robust.
Overall, the results support our hypothesis that training on simple Riemann problems is sufficient to generalize to complex piecewise-constant initial conditions. In particular, the ability of (U)NFV trained only on these analytically tractable Riemann building blocks to generalize reliably to much richer picsewise-constant and real-world configurations turns this seemingly strong assumption into a practical strength rather than limitation. Additional dataset and training details, along with the heuristic exploration and hyperparameter tuning that led the method to work, provided in Appendix D In the next section, we show that NFV also generalizes to experimental highway data, where conservation is often violated and traditional methods typically fail.
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Table Mean and standard deviation of final-time the standard LWR benchmark with Greenshields' flux different CFL ratios, comparing NFV3 with classical finite volume baselines and DG.
CFL
NFV
GD
LF
EO
ENO
WENO
DG
0.2
1.6e Sac
3.8c l.c4
7.60
3.8c lic4
6.0c
6,2c 1ac4
3. 3.0e
0.4
1.3e 2
3.3e 1e
3.3e
6.0e Sae
6.4e 14c
fail
0.6
1.2e sa5
2.1e
1.3e
2.2e
1.5e
1.5e 1h
fail
0.8
1.0e
2.20
2.0e anc
2.3e"ine"
1.6e 220
7.2c 1ac
fail
1.0
9.1e
1nck
1.5e
1.8e
5.6e
9.6e lrck
fail
1.2
1.2e
2.1
1.
2.2
1.5
1.5
fail
Table 4: Improvements of NFV at different scales against numerical methods with fitted flow functions on field data. The reported metrics include LI enror (mean(|u x|)), L2 error (mean((u a)²)), )), and relative (meanu[/ /|/l max(s, (e, m). The larger the input size of NFV, the better the performance. NFV] outperforms all calibrated Godunov fits, despite having the same input size and underlying structure.
Calibrated numerical schemes (Godunov)
NFV (Ours)
Greenshields Triangular
Trapezoidal
Greenberg
Underwood
NFV;
NFV
NFVI0
L1
6.05e-2
2.77c-2 2
2.73e-2 2
2.79e-2
4.98e
2.37e
2.31c-2
2.02e
1.2
1.93e
1.31e
1.30e
1.33e
1.81e
1.23e
1.2le
1.09e
Rel.
5.04e-1
3.83e
3.74e
3.75e
5.45e
3.57e-1
3.5le
2.83e
MODELING LARGE-SCALE EXPERIMENTAL FIELD DATA USING NFV
We apply the proposed NFV method large-scale traffic field data collected on Interstate 24 (I-24) in Tennessee, USA, using the 1-24 MOTION infrastructure (Gloudemans et al. 2023a b). which
enables high-resolution vehicle trajectory collection and constitutes the most extensive publicly available traffic dataset to date. Rather than predicting traffic speed, focus modeling traffic density, which is more directly tied to conservation laws and often exhibits sharp transitions that are challenging capture. Although conservation of mass is not strictly satisfied in highway traffic data due to merges, exits, and incidents, it serves as a strong inductive bias. We show that NFV achieves superior predictive accuracy compared to classical numerical schemes. Moreover, incorporating the PDE structure leads to substantially more stable training, particularly in data-scarce regimes. These findings suggest that our approach can enhance the accuracy and efficiency of traffic simulations, thereby contributing to better-ine informed decision-maker in urban planning and traffic management.
6.1 DATASET AND TRAINING
We evaluate our method the 1-24 MOTION dataset (Gloudeeaas et al. 2023a), which provides nigh-resoluutin vehicle trajectories collected on four-mile stretch of Interstate 24 (mile markers 58.7 to 62.7) near Nashville, Tennessee. The data captured by network high-definition cameras mounted along the highway as part of the 1-24 MOTION infrastructure, leading to intricate trajectory data as illustrated in Figure 9 Vehicle trajectories are reconstructed using a computer vision and data association pipeline (Wang et al. 2022), resulting in high-fidelity, though inherently noisy, field data.
The dataset consists of 10 days of vehicle trajectory data, collected during the morning rush hour (6:00 AM to 10:00 AM) over the 4-mile segment. From the raw trajectory data, we construct spatiotemporal vehicle density fields by aggregating vehicle counts over fixed spatial cells. Details of the data cleaning, processing, and preparation provided in Appendix C. Visualization of the resulting density fields shown in Figure 10
Further training details are available Appendix D
Concretely, all NFV models and tuned finite-volume baselines are trained on the first hour of data from November 29, 2022 using a single boundary cell on each side; the autoregressive prediction horizon is increased from 10 to 100 steps during training, while the learning rate decays from 10 to 10- over roughly 3000-5000 epochs, leading to convergence within 15-30 minutes on a single NVIDIA RTX A5000 GPU.

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Figure 6: Autoregressive prediction of NFVI0 10 (right) compared to the ground truth (left). Full results are shown in Figure Sce Appendix C.3 for how to read the heatmaps.
Table Generalization of NFV against Godunov days of 1-24 data seen during training. As in Table
we report mean and standard deviation of 1, L.2 and relative errors.
L1 error
L2 emrr
Relative error
Godunov
1.56e + 2.02c
3.74c + 8.25e
6.26e + 2.58e
NFV
1.12c + 7.39c -3
2.20c -2 2.59e
3.59c + 7.58c
6.2 RESULTS AND DISCUSSION
We compare NFV to numerical schemes using the flux functions from Appendix B These functions, each defined by few parameters, were calibrated via optimization minimize the Godunov scheme's 's prediction error on the training set. The search ranges were intentionally broad, prioritizing predictive performance over physical plausibility to ensure fair comparison. We chose the Godunov scheme as the representative baseline to compare with since we observed only a marginal performance difference (up to 5%) between Godunov and other FV schemes on this dataset, and the Godunov scheme is known to converge to the entropy solution. We evaluate three NFV variants of increasing capacity: NFV2, NFV3, and NFV1] (training details can be found in Appendix D to assess how well they generalize and capture complex field dynamics.
shows that all NFV models outperform the five tuned Godunov schemes, with performance improving as input size increases. This trend matches what was seen synthetic data (Section ( Despite training on just one hour of data, NFV predicts nearly four hours of traffic evolution autoregressively (Figure 6). While performance degrades in -ut-of-distribution zones (e.g., dark green regions unseen during training), the models still capture key wave patterns with high fidelity. Larger stencils help smooth out noise and improve accuracy, seen in Figure 13
We further evaluate generalization on 7 other days. As shown Figure 14 and Table 5. NFV consistently outperforms the best Godunov scheme the evaluation set, even though both perform similarly on the training day. Indeed, although far from perfect, it able to capture the evolution of free-flow traffic (dark green) with much greater accuracy, allowing of congestion waves (red). NFV scales naturally with capacity: NFV 1o adds only 1728 parameters over NFV? but achieves significantly better accuracy with similar runtime and memory usage, unlike hand-crafted schemes, which significantly grow in complexity (see for example Appendix A
successfully capture the end
7 CONCLUSION
We introduced (U)NFV, a neural network-based framework that extends finite volume methods for hyperbolic conservation laws by learning numerical fluxes over extended spatio-temporal stencils while preserving conservation. (U)NFV achieves high accuracy and efficiency, capturing complex wave dynamics with high fidelity. outperforming classical basclines on standard PDE benchmarks and large-scale field traffic data, Its modular design scales to large spatial and temporal stencils, matching the accuracy of state-of-thh-ar methods such as DG with much lower implementation complexity and significantly faster inference. In parallel work we have established convergence guarantees by controlling error propagation and deriving bounds on network size and training set requirements. These theoretical results will appear in a forthcoming journal publication. Future directions include applying (UNFV to velocity-basee formulations to learn speed-flu ffuu relationships, which aligns with GPS observations and avoids the need for a closed-form velocity PDE. Architecturally. NFV is dimension agnostic, but systematic multi-dimensional empirical study remains for future work.
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REPRODUCIBILITY STATEMENT
To support reproducibility, we provide detailed descriptions of the NFV and UNFV architectures and training objectives in Sections 3 and 4 along with experimental setups Sections and [ The formulations of all classical numerical baselines are presented in A while benchmark equa- tions are introduced Section and expanded Appendix [ Additional implementation details, including model architecture, training procedures, hyperparameters, and dataset processing, provided in Appendix [D] Finally, code, datasets, benchmarks, and trained models are released at hat.://alicccc..cooo/////rarrri
ETHICS STATEMENT
The submission does not have any ethics issues.
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A FINITE VOLUME METHODS
Several finite volume- based numerical schemes are studied in this work. They include the following common classical first-order schemes:
The Godunov method (Godunov 1959a):
min f [u7-1>"(] max f if u u.?
if uup S u
Vi,n n P
The Lax-Friedrichs method LLax 1954):
Ax
Vi, n
P (f(u)) + f(uf-1))
x lu, u.-1/.
2 At
The Engquist-Osher method (Engquist and Osher 1981):
1 (f(ul) + f(uf
Vi, n F 21
If
Additionally, higher-order schemes such as the Essentially Non-Oscillatory (ENO) method (Shu 1999) and the Weighted Essentially Non-Oscillatory (WENO) method (Liu et al., 1994) are considered. The main idea in these methods is that by considering stencils, expect increase the accuracy of approximation of the solution.
For the ENO scheme, consider the semi-discrete form of
0yu: Ar (Fit1/2-Fi-1p 1/21/2).
(6)
Using the Lax-Friedrichs Splitting technique, have
1 f(u) f+) + (u), (w)) F(u) ) =au),
where = max) is the maximum wave speed. The key point in the ENO scheme is the high-order upwind interpolation of ft and f based the smoothest stencils. For instance, for the 2 2-stencil ENO scheme, the procedure is as follows:
1. Evaluate the smoothness indicators:
== - f-1l, 8- 1f - ft
2. Select the stencil that minimizes the smoothness indicator:
If 84 8_, choose the stencil {f. f
Otherwise, choose the stencil (
3. Perform linear interpolation to compute the numerical flux:
where 8+ is the difference between the selected stencil elements.
A similar approach is applied to compute l using the right-biased stencil.
The final numerical flux the interface is obtained by combining the positive and negative parts:
= f
In this work, have used 3-stencil scheme for ENO.
The WENO scheme follows the same idea ENO by using specific weights in defining ft rather than explicit conditions. In this work, the 5-stencil WENO scheme.
1
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B VARIANTS OF LWR
We consider six different LWR PDEs variants, each consisting of a different fundamental diagram, illustrated Figure All of the considered flows are concave continuous mappings from [0, Pmaxl to R4. where Pmax is the maximum density, with the exception of the Greenberg flow whose domain is (0, Pmax). The critical density P. denotes the density at which the flow is maximized, i.e. Pr: arg maxxe((, pe[00,pmsa] [(p). The following introduces the six flow models we consider in this work, each time detailing the flow's parameters, parameter values we use Section ( parentheses), and the flow's definition. Note that we consider normalized parameter values lying between 0 and 1 for the most part.
Greenshields
Parameters: free-flow speed Vmax (1 m/s), maximum density Pmax (1 veh/m).
f(p) Umaxp (1-
P
Pmax
Triangular (symmetrical) Parameters: free-flow speed Umax (1 m/s), critical density Pe (0.5 veh/m), maximum density Pmax (1 veh/m), wave propagation speed (-1 m/s).
Umaxp
if p Pe
[())
wpp Pmax)
if p Pe
non-symmetric variant of the Triangular flow, with parameters Umax m/s, Pr: = 1/3 veh/m, and w = -1 m/s.
Triangular (skewed)
free-flow speed Umax (1 m/s), first density cusp Pi (0.2 veh/m), second density cusp 02 (0.8 veh/m), maximum density Pmax (1 veh/m), wave propagation speed (-1.5 m/s).
Trapezoidal Parameters:
UmaxP
p < P1
f(p)
(wop2 -Pmxx) UmaxP1)
P-P1 P1
+ UmaxP1
if S S P2
P2 P
w(p Pmaxx
if p >
Greenberg Parameters: maximum density Praxx (1 veh/m), coefficient co (2).
f(p) cop (Pmax/p)
Underwood Parameters: maximum density Prax (1 veh/m), coefficients eı (0.25) and 02 (1).
f(p) = expexp (1 C2P)
Example solutions of the Greenshields LWR are shown on various initial conditions Figure
C I-24 EXPERIMENTAL DATASET: DENSITY EXTRACTION, TRAINING, AND EVALUATION
C.I DENSITY FIELDS EXTRACTION FROM I-24 MOTION DATASET
1-24 MOTION is a large-scale traffic monitoring system installed along a section of Interstate 24 near Nashville, Tennessee. It uses dense network of high-resolution cameras and computer vision algorithms to capture detailed, real-time vehicle trajectories across multiple lanes and miles of highway. The data collection network and resulting trajectory data are illustrated in Figure
For our experiments, we use the INCEPTION datase ( Gloudemans et al. 2023a) from 1-24 MOTION, consisting of ten days of data, each covering the morning rush hour (6:00 AM to 10:00 AM). The dataset for each day comprises 15-20 GB stored as single JSON file. We first split each file into
Availlabe 124mot .rg as part of the INCEPTION data release.
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f(u)
fu)
f(u)
0.2
0.4
0.4
0.1
0.2
0.2
u
u
0
0.5
0
0.5
0.5
()) Greenshields
(b) Triangular
[c) Triangular' 2 (Skewed)
f
(u)
fu)
f f())
0.2
0.6
0.2
0.4
0.1
0.1
0.2
u
0
0.5
0
0.5
0.5
(d) Trapezoidal
(c) Greenberg
(f) Underwood
Figure 7: Flow models for LWR. We consider six different variants of the LWR PDE with the flows illustrated here, each mapping road density (veh/m) to traffic flow (veh/s).
manageable 1 GB chunks and parse them efficiently using s simdijson (Langdale and Lemire 2019) which enables extraction density fields in approximately 3-5 minutes per 20 GB file.
To construct the density fields, we discretize the spatiotemporal domain into cells of size 0.02 miles (s:32 meters) in space and 0.1 seconds time, aggregating data across all four lanes. Vehicle counts in each cell are normalized to obtain densities in vehicles per kilometer per lane. To reduce noise, we average over 100 consecutive time steps (i.e., 10 seconds) and over adjacent spatial cells (i.e.. 0.04 miles or =6 meters). This results in grid of 100 spatial cells (4 miles / 0.04 miles) and approximately 1440 time steps hours / 10 seconds). We clip the first and segments of each day to exclude low-density, free-flow regimes with incomplete data, retaining 1300 time steps per day depending on data quality. To avoid extreme outliers, we cap densities at 140 vehicles/km/lane. For all training and evaluation purposes, then normalize densities so that the maximum density is 1.
Due to occasional sensor failures, such as malfunctioning camera poles or occlusions by bridges, there seven spatial locations with missing data. We fill these gaps by linear interpolation between the adjacent upstream and downstream cells.
1
t
t
t
W
0
1
u(t
0.
x
u(t
0, .x)
2
utt
x))
0
0.5
0
0.5
0.5
(a) A Shock wave.
(b) Rarefaction wave.
(c) Eval IC.
Figure 8: Exact solution for two Riemann problems (left, middle) and piccewise-constane initial condition (right) from the evaluation set.
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Figure 9: 1-24 MOTION illustration. High-definition camera poles are mounted along portion 1-24 regular intervals. This generates massive amounts of vidco data, which is processed through a software stack. The resulting data for a single day is shown in the time-space diagram, displaying thousands of individual vehicle trajectories color-coded by speed (red for low speeds, green for high speeds), illustrating the complexity of the dataset.
Figure 10 shows the density fields extracted from 1-24 MOTION data. Higher densities (in red) correspond to stop-and-go waves and congestion, while lower densities (green) correspond to free-flow traffic. The processing code and resulting data are available in our codebase.
We exclude data from November 24 and 25, 2022, from our analysis, as both days correspond to holiday periods with purely free-flow, low-traffic conditions and no observable stop-and-go waves. These days are therefore not relevant to our study, which focuses on modeling traffic dynamics in the presence of congestion. The remaining days still include sufficient free-flow segments evaluate model robustness in those regimes. Nevertheless, we include the excluded days in the released dataset for completeness.
C.2 BOUNDARY CONDITIONS
For both training and evaluation NFV on the 1-24 dataset, we initialize the model using single time step of real data and provide one boundary cell at each end of the road, using the corresponding real values. While it possible to use additional ground truth data improve accuracy, we deliberately restrict ourselves for two main reasons: (1) to allow fair comparison with the Godunov scheme, which uses single boundary cell per side, and (2) to reflect realistic deployment scenarios, where boundary densities might only be measured at few fixed points (e.g., at the road extremities), or predicted using a separate model.
For models that require wider input steneil (c.g., those larger than NFV.), pad the boundaries by duplicating the available single-cell values. This ensures that all models, no matter their size: Godunov, NFV, NFVII 10- receive the same amount of boundary information. Figure 11 illustrates the boundary setup in both cases, showing which values are provided as input and which are left to be predicted.
Finally, we emphasize that initial and boundary conditions are not included when computing metrics, whether in the training loss at evaluation.
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C.3 READING THE HEATMAPS
This section provides a brief explanation and intuition for interpreting the heatmaps displaying 1-24 MOTION data. The horizontal axis represents time, increasing from left to right, while the vertical axis represents space along the road, increasing from bottom to top. The color encodes traffic density, normalized between 0 and according the colormap shown in Figure 12 where green indicates low density traffic (free flow) and red indicates high density traffic (congestion). Unless otherwise specified, only the model predictions are shown, while initial and boundary conditions are omitted for clarity. Stop-and-go waves appear as high-density (red) bands that propagate upstream, i.e., move backward through traffic.
C.4 PREDICTIONS
Predictions on the training day from Section 6.2 are displayed in Figure 13 Predictions on evaluation days are displayed in Figure 14
D EXPERIMENT DETAILS
D.1 MODEL ARCHITECTURE
The model is applied locally on each cell to estimate the corresponding numerical flux. It is implemented as wwo-dimensional CNN. The first layer convolutional layers with 15 channels and kernel size 1. Using a CNN enables efficient vectorized computation over all stencils, which is equivalent sliding fully connected network along the input but significantly faster. Each time step represented as a separate input channel, for a total of 6 input channels. Note that when single channel providing the estimated flux. Each NFV model consists of hidden layers of width 15, totaling 1 1105 + 16(ab b + 1) parameters for NFV e- This remains quite small, with around 1200 trainable parameters for the smallest variant, which intentionally chosen the smallest architecture that achieves competitive performance while consistently outperforming first-order FV baselines across our benchmarks.
kernel of size followed by five
1, one-dimensional CNN can be used. The output is
We also found that activation functions have a modest effect: ELU activations perform slightly better on smooth flow functions (Greenshields, Greenberg, Underwood), while ReLU is preferable for piecewise-linee linear flows (Triangular, Trapezoidal). However, the difference in performance is minor.
D.2 TRAINING ON SYNTIETIC DATA
For the LWR model, training is performed autoregressively: the model predicts future time steps by feeding its own outputs as inputs. We use 2048 randomly sampled Riemann problems (P1,P2) for training, which proved more effective than uniformly spaced samples. To encourage generalization, small perturbations are added the discontinuity location. Empirically, increasing the number of training Riemann problems beyond this scale did not yield noticeable accuracy gains and mainly increased training time, indicating that performance in this regime
limited by data size.
The discretization parameters are At = 5- 10-3, Ar = 10-2, 100 space cells, and up to 250 time steps. Evaluation done on 100 test cases generated with a finer grid and the Lax -aof algorithm: At = 10 Ar 10-3, with 200 cells and 1000 time steps. A CFL ratio of 0.5 (e.g. dr 10-3 and = 10-4) was robust across different flow functions. Higher CFLs sometimes work but were less reliable. For instance, CFL of 1.0 (e.g. dæ d!) is effective for the Greenshield flux but leads to poor performance on the Triangular flux.
The prediction horizon is progressively increased from 10 to 250 steps during training. Most progress occurs at short horizons (10 steps already outperform Godunov on average), while longer horizons provide additional fine-tuning and stability. The learning schedule that proved robust is summarized below:
For unsupervised experiments, compute the weak loss function using test functions consisting of 250 randomly sampled compactly supported polynomials of degree 50. This proved to work well across PDEs, and details test function generation are provided in the released codebase.
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Empirically, observed that performance is largely insensitive the specific random draws, number, degree, or family of test functions once these values are moderately large, and that similar results obtained with trigonometric test functions; degradation only appears for very small numbers or degrees (around 1-2), where the weak loss becomes poorly conditioned, so we found that the chosen setting offers stable yet memory-efficien default.
Stage
Training steps
Learning rate
72
R2
1
10,000
1 10
10
10
2
20,000
10-5
50
50
3
20,000
5-10 10
100
100
4
20,000
1.10-
200
200
Here, (Ma,, denotes the size of the space-time window predicted autoregressively. Most learning occurs during the first stage, with the later stages serving as progressive fine-tuning. Training uses the Adam optimizer with a decaying learning rate, from 10 4 to 10 1 and the largest batch size that fits in memory (ideally the entire dataset). Training RTX A5000 GPU takes about 30 minutes.
D.3 TRAINING ON EXPERIMENTAL DATA
All models and fitted finite volume schemes are trained on the first hour of data from November 29, 2022. and evaluated on the full morning period (nearly four hours) and the remaining days of data, To ensure fairness and reflect practical deployment constraints, each model only receives a single boundary cell on each side, as described Appendix C.2 from additional context.
even though larger models could benefit
Each NFV model consists of 6 hidden layers of width 15, totaling 1105+ 16- (a-b+1 1) parameters for NFV* Training takes 15-30 minutes on an RTX A5000 GPU. The prediction horizon increases from 10 to 100 steps during training, while the learning rate decays from 10-3 to 10-1 over 3000-5000 epochs depending on model size, until convergence.
D.4 HARDWARE AND RUNTIME
single NVIDIA A5000 GPU with 24GB of VRAM. The codebase (including (U)NFV and baselines) is fully vectorized so that all solutions are computed in parallel. Among classical schemes, aux-Friedrichs is fastest due its simplicity. Godunov, Engquist-Osher and (U)NFV are 2-3x slower, ENO and WENO are 6x slower, and DG is up to 20x slower, with significantly larger memory requirements. Equivalently, NFV runs within small constant factor of Godunov, is roughly twice as fast as ENO and WENO, and more than an order of magnitude faster than DG our benchmarks, while still benefiting from GPU batching across hundreds of solutions where DG must be evaluated in smaller batches. These relative runtimes are hardware dependent but give representative picture.
All experiments
Training is also relatively fast GPU. The model generally surpasses the Godunov baseline after only a few minutes and reaches most of its final performance within 15 minutes. In most runs extended training to one hour more, though the remaining time typically yields only minor fine-tuning. Figure 15 illustrats a typical training curve. To quantify progress, we also compute a winrate metric, defined as the percentage of evaluation initial conditions
which the model achieves lower La error than a baseline such as Godunov. Under this metric, the model usually remains at 0% for the first few minutes, then rapidly increases above 95%, and often reaches 100% winrate against Godunov within the first 15 minutes of training. A typical winrate curve is also depicted in Figure 15
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Published as a conference paper at ICLR 2026
(a) Nov 21, 2022
(h) Now 22, 2022
(c) Nov 23, 2022
Nov 28, 2022
Nov 29,
30,
(g) Dec 01, 2022
(b) Dec 02, 2022
Figure 10: Time-space diagrams of car trajectories extracted from the video, colour-coded by density, for different dates. Sec Appendix C.3 for how read the hcatmaps.
(a) Nov 2022 boundary conditions for NFV)
Nov 2022 boundary conditions for NFV
Figure 11: Boundary conditions used by NFV during training and evaluation on the 1-24 dataset. The figures show input provided
model: initial condition at f on the left, and boundary conditions at æ (bottom) and x = Tmax (top). The model must then predict the interior (i.e., the region shown in Figure 10) autoregressively: it uses its own output time f to predict the state at time t + dt, without receiving any additional data beyond the fixed boundaries. Note that both figures the same underlying data; for NFV1. 10- the boundary values are duplicated provide the required input padding.
Density
Figure 12: Colorbar showing density scale for all 1-24 data heatmaps.
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Published as a conference paper at ICLR 2026
FV li:: Greenshields
(b) FV Triangular
(0) FV fit: Trapezoidal
(d) PV fit: Greenberg
(0) FV fir: Underwood
Ar:: NFV.
(g) FV fit: NFV5
(h) FV fit: NFV1
① Ground
Figure 13: Predictions of FV methods and trained NFV. Corresponding metrics are reported in Table Among the FV methods, only the Triangular, Trapezoidal, and Greenberg flows provide a reasonable fit to the I-24 MOTION data, In contrast, NFV models show increasing predictive accuracy with model complexity. For example, NFV1] 10 captures significantly more stop-and-go waves (in red) than NFV; or NFV as well as fast low-density waves (in green), cnabling it correctly predict the early dissipation of the final two waves. However, it exhibits oscillations toward the end of the prediction window, likely due to limited generalization caused by the scarcity of low-density (dark green) patterns in the training data; neverthcless, the primary objective when modeling experimental data is to accurately capture the evolution of congestion waves, whereas free-flow traffic is of lesser interest. All models were trained only the first 25% of the ground truth sequence, and the predictions are generated fully autoregressively. See Appendix for how to read the heatmaps.
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Published as a conference paper at ICLR 2026
Prediction on training day
(a) Ground truth (Nov 29, 2022)
(b) NFV prediction
(c) Godunov prodiction
Generalization on unseen days
(d) Ground truhh (Nov 2022)
NFV prediction
(f) Godunov prediction
(g) Ground truth (Nov 22, 2022)
(h) NFV prediction
(i) Godunov prediction
(i Groand truth (Nov 23, 2022)
(k) NFV prediction
① Gadunov prediction
(m) Ground truth (Nov 28, 2022)
NFV!! 20 prediction
(0) Gadunav prediction
(p) Ground truch (Nov 30, 2022)
(g) NFV prodiction
(0) Godunov prediction
(5) Ground truth (Dee 01, 2022)
(1 NEV prodiction
(u) Godunov prediction
(v)
2022)
(w) NFV prediction
(x) Gaulanav prediction
Figure 14: Predictions of best and trained NFVII. Godunov is derived by fitting flow function on the prediction and comparing it against the ground truth; we keep the fitted Trapezoidal flow as performed best (see Figure 13 and Table 4. Both the Godunov fit and the NFV!! training are realized using the same data, namely the first hour of Nov 29, 2022 data (i.e.. the first 25% of subligure (a)). This means that the remainder the data on that day (row 1), as well the prediction all subsequent days (rows 2-8) are generalization on data that was never seen before by either models. See Appendix C.3 for how to read the heatmaps.
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Published as a conference paper at ICLR 2026
100
10-1
Model L2 error Godunov baseline
Average L2 error
Winrate (%)
10
50
10
10
15
10 Training time (min)
15
Training time (min)
Figure 15: Training dynamics. Left: Average Le error of the model and of Godunov baseline. Right: Winrate of the model against the Godunov baseline. the metrics computed periodically on evaluation dataset 100 random complex initial conditions, with prediction over 1000 timesteps.
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Reviews:
[{"review_id": "qJbdJXGzR5", "reviewer": "Reviewer_j2sz", "summary": "This paper presents a neural method to solve hyperbolic PDEs. The core method is called neural finite volume (NFV), and is inspired by the finite volume methods that attempt to satisfy local conservation of the appropriate variables. The unsupervised variant of this method uses the basic concept of the update rule of a generic finite volume method. Thus, the loss function is the squared norm of the residual that is obtained from this update rule. This loss is then minimized to train the network. The method is applied on two benchmark problems, namely, the traffic flow equation (many variants) and the Burgers' equation, with competitive results against classical finite volume methods. Finally, a supervised variant of NFV is also presented, trained on a real dataset.", "questions": "* My understanding is that the proposed method is intended to solve only one instance of an equation, correct?\n\n* It seems to me that, we need to solve an optimization problem at every time step. Is that correct? In that case, this method is not very competitive in terms of time to solve. What do the authors think?\n\n* My understanding is that, to reach $u^{\\Delta t}$ from $u^0$, one uses the neural network to predict the fluxes, and then update the solution, and iteratively solve the optimization for this one time step, and obtains a good approximation of $u^{\\Delta t}$ at the end. And then, one proceeds to calculate $u^{2\\Delta t}$. Is that correct? I think, a flow chart would be a good addition.\n\n* The authors remark that it is very easy to implement this method compared to the classical methods. Could the authors elaborate on this? How easy is this method? In what way? Usually, in a conventional code, the important pieces are the correct differentiation rules / stencils, communication between values in the volume and the values on the facets, and implementing the special adjustments at the boundaries, among many other aspects. How does NFV fare on these considerations?\n\n* Also, by the virtue of choice of the example PDEs, I am assuming that the authors did not need to calculate the spatial derivatives of the primal variables during the training process. That is, both the $t-$ and the $x-$ derivatives are integrated away, and the loss functional $L_w$ does not contain any spatial derivatives of the function $u$. Is that correct?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* The paper is well-written, with ideas clearly conveyed.\n\n* The method is shown to be yielding lower errors compared to some of the classical methods. The chosen equations are in one dimension, and have one unknown. But they are usually the standard places to start.\n\n* Numerical convergence results are presented, showing the behavior of the errors with respect to the time-step size.", "weaknesses": "* One of the flaws in this paper is that the neural network design is not discussed enough, or relegated to the appendix. While it is understandable that, a significant portion of this design is heuristic, it is still important to make an attempt to synthesize that knowledge, especially when one is trying to solve scientific computing problems.\n\n\n* While the comparisons with the classical methods are essential, it is also very important to compare a new method with existing methods that are not classical. Currently, no comparisons are made with other existing neural network based methods, especially in the unsupervised regime.\n\n* The paper does not flesh out the limitations of this method. What happens when a rarefaction happens? Does the method select the correct weak solution? The authors do mention that imposing the $L_2$ norm loss does not guarantee the entropy solution, but remark that most of the numerical experiments show that the method selects of the entropy solution. Such a remark only pertains to the few cases considered in the current work. \n\n$\\textbf{Minor}$\n\n* No equation number on some equations, e.g., line 236 ($L_w$).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xh8I8kFnOd", "reviewer": "Reviewer_pXtN", "summary": "The paper proposes (U)NFV, a neural finite volume (FV) framework for 1D hyperbolic conservation laws, that learns numerical fluxes on extended space-time stencils, while keeping the classic FV conservation update, so mass is conserved by construction. It comes in two flavors: NFV (supervised) trained on solution data, and UNFV (unsupervised), trained with a weak form residual loss, targeting entropy solutions without labeled data. The main contributions are: (1)  (U)NFV is a conservation preserving neural FV generalization, (2) supervised and weak form unsupervised training options, (3) strong accuracy vs. FV/ENO/WENO and DG-level performance with far simpler implementation, (4) successful application to noisy, non-ideal field data, and (5) empirical convergence with notes on parallel theoretical guarantees to appear. In summary, it is a practical, accurate, and physics respecting route to data driven solvers for hyperbolic conservation law, which is useful in diverse areas, from scientific computing to real world systems like traffic.", "questions": "Suggestions: (i) State, or at least sketch, the promised convergence guarantees for (U)NFV. Add an entropy consistency discussion and a discrete mass/entropy error monitor across rollouts, (ii) For UNFV, analyze identifiability of the weak loss, and report sensitivity to the number/degree/family of test functions.\nRun a CFL sweep and report stability/accuracy curves (CFL ∈ [0.2,1.2]) across fluxes, and quantify failure modes. (iii) Add a 2D case,  as a short appendix experiment, to show that the method survives beyond 1D scalars.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "(i) The method keeps the classic FV update, so mass conservation is guaranteed, while learning the numerical flux on extended space–time stencils, (ii) It supports both supervised NFV and unsupervised UNFV, (iii) (U)NFV reports up to 10× lower error than classical FV, and matches DG level accuracy without DG’s complexity. On I-24 MOTION field data, NFV outperforms calibrated Godunov baselines, and handles noisy conditions. Empirical refinement studies suggest convergence, and models trained on simple Riemann problems generalize to more complex initial conditions. A small CNN makes large stencils practical and fast, avoiding the hand-crafted complexity of high-order schemes.", "weaknesses": "(i) All experiments are on 1D, first-order scalar conservation laws. Multi-dimensional problems and systems, are left for future work, and acknowledged to bring extra stability/complexity challenges, (ii) The UNFV weak form loss does not guarantee convergence to the entropy solution, and the success is reported empirically only. Formal guarantees are deferred, and are not contained here, (iii) Training/rollouts are most reliable at CFL ≈ 0.5; higher CFLs sometimes work but are less reliable, and CFL=1.0 degrades on Triangular flux, which is stricter than typical Godunov stability limits.  NFV solutions can show spurious oscillations, even when Godunov is monotone/TVD, both on synthetic PDE tests and in field-data rollouts.  Supervised models are trained predominantly on Riemann problems, which is a strong but narrow distributional assumption. The weak loss uses 250 random polynomials of degree 50, and performance may be sensitive to such design choices.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dOkXriguA0", "reviewer": "Reviewer_zh6x", "summary": "This paper introduces Neural Finite Volume, a neural network architecture that generalizes classical finite volume methods for solving hyperbolic conservation laws. The key innovation is learning numerical flux approximations over extended spatiotemporal stencils while preserving the conservation structure of traditional FV methods. The authors develop both supervised and unsupervised variants, demonstrating up to 10× improvement over Godunov's method on benchmark PDEs and showing practical applicability to real highway traffic data from the I-24 MOTION dataset.", "questions": "1. How would NFV handle non-periodic boundary conditions (Dirichlet, Neumann, open boundaries)?\n2. How does the method compare to some of the other NN-based methods as mentioned above?\n3. Is it possible to conduct ablation studies on architecture choice, stencil size selection, or the training data size requirements?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed NFV framework provides a method to incorporate neural networks into FV methods while maintaining conservation.\n2. The application to real traffic data seems valuable and shows practicality.\n3. The comparisons with classical methods provide useful context\n4. Table 1 shows consistent improvements across multiple PDEs", "weaknesses": "1. the method is restricted to 1D conservation laws. Although the authors claim that \"NFV architecture is, in principle, extendable to higher dimensions\", it has not been demonstrated in the paper and may introduce significant challenges.\n2. The comparisons are limited to classical numerical methods. It would be significantly beneficial if the authors could compare their methods to some of the recently proposed neural network based method, (e.g. https://proceedings.mlr.press/v235/chen24ad.html, https://arxiv.org/abs/2410.22193, https://proceedings.mlr.press/v202/muller23b.html). If the comparison is difficult, maybe the authors can comment more on how their method differs from these methods and how their method may be advantageous?\n3. When does NFV fail? The paper shows it works well on the tested cases but doesn't explore boundary conditions beyond periodic, complex geometries, or cases with strong shocks.\n4. While Table 3 shows NFV variants with increasing complexity, the paper doesn't discuss the training time or data requirements. How many Riemann problems are needed? How long does training take compared to one-time use of a classical method?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oky8KtMsP8", "reviewer": "Reviewer_Z2F1", "summary": "This manuscript proposes the U-NFV framework, which employs neural fields to solve PDEs under both supervised and unsupervised settings. Existing neural fields depend on labeled data (supervised) or require heavy reliance on physics-informed losses (unsupervised). U-NFV introduces a variational loss functional to allow for both training settings, a neural functional operator that maps input coordinates to PDE residuals and boundary losses, and a hybrid optimization strategy which combines direct residual minimization (unsupervised) with data-driven regression (supervised). Experimental results show that U-NFV achieves SOTA accuracy across both data-rich and data-sparse settings, and exhibits continuous resolution generalization.", "questions": "- Please refer to the \"Weaknesses\" section above.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **Technical novelty**: The proposed general neural variational operator is compatible with both data-driven and physics-based objectives.\n- **Extensive experiments**: The proposed method is tested across both linear and nonlinear PDEs, multiple spatial dimensions, and diverse boundary conditions. The experimental results show SOTA-comparable accuracy. \n- **Resolution generalization**: Neural field representations enable solving PDEs at unseen spatial resolutions without retraining.\n- **Mathematical backbone**: The proposed method links to the Ritz-Galerkin method and classical variational formulations.", "weaknesses": "- **No efficiency comparisons**: Metrics like training time, inference time, and memory consumption are missing. \n- **Scalability**: Experiments are limited to PDEs with moderate grid sizes. The framework’s computational feasibility for large-scale 3D problems is not discussed.\n- **Uniform sampling**: Uniform point sampling is used for both loss variants. Whether the method allows for adaptive residual sampling is unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)