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A neural characteristic mapping method: Lagrangian PINNs based on flow maps for transport-dominated problems
Martin Campos Pinto, Victor Fournet, Emmanuel Franck, Philipp Krah, Victor Michel-Dansac
TL;DR
Transport-dominated solutions develop fine structure that is difficult for grid methods and PINNs representing the solution directly. The paper learns the flow map with composed PINN submaps and reports advantages over Eulerian PINNs, including on advection and two-stream instability.
Problem
Transported fields develop finer spatial structure over time, challenging grid methods and PINNs that represent the evolving solution directly.
Source: §1 · p. 1, §3.2 · p. 5Method
The method learns the flow map rather than the transported field, composes submaps learned on separate time intervals, and uses Picard iteration for velocities coupled through field equations.
Source: §2.3 · p. 4, §5 · p. 23Results
Where both methods ran, Lagrangian PINNs clearly outperformed Eulerian PINNs, by more than an order of magnitude on advection, and reproduced the two-stream instability where Eulerian PINNs failed.
Source: §5 · p. 23Takeaways & Limitations
In the tested problems, submap composition carries growing solution complexity through successive maps rather than requiring one network to represent it.
Source: §2.3 · p. 4, §5 · p. 23In the low-dimensional settings considered, the method is not cost-competitive with grid-based solvers, and solution-dependent velocities make each new submap cost grow linearly with the number of previous submaps.
Source: §5 · p. 23
Abstract
from arXiv · showhide
We propose a novel numerical method for transport-dominated problems using Physics-Informed Neural Networks (PINNs). In such problems, the solution can develop large gradients and fine structures over time. This is particularly challenging for classical PINNs, which represent the solution itself. In our approach, rather than the solution, we choose to approximate the flow map of the equation. The solution is then recovered by pulling back the exact initial condition, so that its fine structures are produced by the map rather than represented by the network. Our method is constructed by approximating the flow map of the equation with multiple PINNs, one per time subinterval, and composing them using the semigroup property of the flow, so that each network only has to represent a map close to the identity. To highlight the advantages of our method compared to classical PINNs or classical numerical schemes, we present numerical results on the linear advection equation, the incompressible Euler equation in vorticity formulation, the Vlasov-Poisson equation, and finally a drift-kinetic equation.
1 Introduction
Transport-dominated problems develop fine scales that challenge both grid-based solvers and PINNs trained directly on the transported field. This work learns the flow map instead, composing submaps and evaluating PINN and Picard-based methods across several transport problems.
1 Introduction
- The method targets transport-dominated problems where stretching and folding generate fine scales that grid methods must resolve and direct PINNs struggle to represent.PINNs determine parameters by minimizing the governing-equation residual without reference data, but a single network must represent a field whose spatial complexity grows over time.Source: §1 · p. 1
State of the art on characteristic mapping methods
- Grid-based characteristic mapping spans transport, kinetic, Euler, and magnetohydrodynamic models, but solution accuracy is ultimately limited by map interpolation [29, 22, 23, 25, 41, 40, 42, 43, 45, 46, 44].Neural flow-map approaches include PINN learning for one-dimensional Vlasov-Poisson, while the present approach determines the map from the governing-equation residual.Source: State of the art on characteristic… · p. 2
Scope and aim of the paper
- The paper aims to learn transport solutions through flow maps, using separate networks for each time subinterval so that each map remains close to identity.The accumulated solution complexity is transferred from individual networks to the composition of successive maps.Source: Scope and aim of the paper · p. 2
Contribution
- The contributions include a framework for composing neural submaps, sequential training with time-independent per-problem cost for solution-independent velocities, and a Picard iteration for nonlinear velocity coupling.The methods are assessed on several transport-dominated problems against classical methods.Source: Contribution · p. 2
2 Flow Maps and the Semigroup Property
The section defines the flow map for a transport equation whose velocity may depend nonlinearly and nonlocally on the solution. Its semigroup property decomposes the backward map into submaps, distributing the growing deformation across shorter time intervals.
2.1 Setting
- The setting allows a velocity field that depends nonlinearly and nonlocally on the transported solution.When the velocity depends on the solution, the flow map and solution must be solved simultaneously.Source: §2.1 · p. 3, §2.2 · p. 3
2.2 Flow maps
- The flow map traces a point at time t backward to its position at an earlier time, and the solution can be recovered from the map and the initial condition.When the velocity is divergence-free, the exact flow preserves volume; for Hamiltonian velocities, it is also symplectic.Source: §2.2 · pp. 3–4
2.3 Flow decomposition into submaps
- Repeated application of the semigroup property factorizes the backward map into submaps, each depending only on the dynamics over its own interval.The solution can then be reconstructed using the initial density and the submaps' Jacobian determinants.Source: §2.3 · p. 4
- A submap covers only one short time interval, so it remains a small perturbation of the identity even as the full flow stretches and folds the domain.This distributes the total map's increasing complexity across the submaps rather than requiring one map to carry it all.Source: §2.3 · p. 4
3 Approximating Flow Maps with Physics-Informed Neural Networks (PINNs)
The section contrasts solution-based Eulerian PINNs with Lagrangian PINNs that learn the flow map and recover the solution by pulling back the initial condition. It then develops submap composition, field coupling, and network architectures for this approach.
3.2 Eulerian PINNs
- Classical Eulerian PINNs fit the solution itself, but their all-times optimization is stiff and increasingly fine solution structures become harder for a smooth network to represent.The text notes that quasi-second-order optimizers may be needed to reach acceptable accuracy.Source: §3.2 · p. 5
3.3 Lagrangian PINNs
- A Lagrangian PINN approximates the backward flow map and reconstructs the solution by pulling back the initial condition, shifting the learned representation from the transported field to its characteristics.The map is trained by minimizing the residual of the Cauchy problem it satisfies.Source: §3.3 · p. 6
- Lagrangian PINNs use the exact initial density and recover fine structures through composition with the learned map, while bijective maps preserve mass exactly.If the network is volume-preserving, the solution also shares the initial density's range and conserves all Lp norms exactly.Source: §3.3 · p. 6
3.4 Lagrangian PINNs with a semigroup property
- A single full-interval map becomes harder to represent as the flow stretches and folds the domain, motivating separate networks for successive subintervals.Increasing the number of submaps makes each closer to the identity, but requires sequential training and increases evaluation cost.Source: §3.3 · p. 6, §3.4 · pp. 6–7
- Sequential submap training optimizes one network at a time with earlier networks frozen, avoiding a fitted density at interval boundaries but requiring k + 1 network evaluations to evaluate the composed density.Each interval uses the identity as its initial condition, and the composition carries continuity in time.Source: §3.4 · p. 7
3.5 Coupling PINNs to the field equation
- When velocity depends on the solution, the method alternates flow optimization with velocity updates using a Picard iteration; prescribed velocities need no repeated field update.At a fixed point, the iteration satisfies both the corresponding loss and the field equation defining the velocity.Source: §3.5 · pp. 7–8
3.6 Architectures of the neural networks approximating the flow maps
- The flow-map architectures enforce structural conditions such as identity initialization and periodicity, while symplectic networks preserve unit Jacobian and exactly conserve Lp norms.Perturbation-of-identity networks impose periodicity through an input embedding; symplectic networks are used when the velocity is Hamiltonian.Source: §3.6 · pp. 8–9
4 Results
The results section reports tests on advection-diffusion cases, incompressible Euler equations, and kinetic equations.
4 Results
- The numerical studies progress from advection-diffusion test cases to incompressible Euler and kinetic equations.The kinetic equations include sections on kinetic equations generally and drift-kinetic equations.Source: §4 · p. 9
4.1 Advection-diffusion equations
The advection tests compare direct solution approximation with flow-map reconstruction, which transports sharp initial features without requiring the network to represent them. With diffusion, a co-moving field carries the remaining evolution, and the Lagrangian approach is more accurate than the Eulerian baseline across the tested architectures.
4.1.1 Simple advection
- The Lagrangian PINN transports the sharp smiley contours without smearing, whereas the Eulerian PINN loses the two eyes by t = 0.8 despite a larger training budget.Figure 3 compares the exact solution and both predictions at t = 0 and t = 0.8.Source: §4.1.1 · p. 11, Figure 3 · p. 11
- For pure advection, the flow-map approach is more than one order of magnitude more precise than the Eulerian PINN, and using submaps improves accuracy over one map.Figure 4a compares mean pointwise errors for both approaches.Source: §4.1.1 · p. 11
4.1.2 Simple advection with diffusion
- With diffusion, the flow map handles advection while an Eulerian field in co-moving coordinates carries the remaining evolution.The map handles transport; the co-moving field describes the slow diffusion and changes smoothly in time.Source: §4.1.2 · p. 12
- With diffusion, Lagrangian PINNs have mean density errors of 1.6·10−3–1.7·10−2, while the Eulerian PINN does not get below 2.0·10−2.The best configurations are about one order of magnitude apart; Figure 4b plots mean flow-map and density errors versus network depth.Source: Figure 4 · p. 12, §4.1.2 · p. 13
- At t = 0.8, the best Lagrangian PINN has relative L2 error 2.8 · 10−2, versus 0.68 for the Eulerian PINN.The Lagrangian prediction retains the slotted-disk notch and cross arms as they begin to blur; its error decreases as diffusion smooths the solution.Source: §4.1.2 · p. 13
4.2 Incompressible Euler equations
The incompressible Euler tests assess Lagrangian PINNs on evolving vorticity fields, including conservation of enstrophy and energy. Composed submaps substantially improve four-modes conservation errors and recover fine vortex-merger spectra beyond the resolution of the coarse spectral solve.
4.2 Incompressible Euler equations
- Enstrophy and energy are exactly conserved by the continuous Euler flow, so their conservation errors quantify numerical or approximation error.Both quantities are monitored in the two test cases.Source: §4.2 · p. 13
4.2.1 Four-modes test case
- In the four-modes test, Nt = 8 submaps reduce enstrophy error to 3.1 · 10−4, versus 2.9 · 10−2 for the Eulerian PINN and 1.6 for a single map.The characteristic mapping method conserves both quantities more accurately, using 105 submaps rather than 8; Table 1 reports runtime, conservation errors, and submap counts.Source: §4.2.1 · p. 15, Table 1 · p. 15
4.2.2 Vortex merger and subgrid resolution property
- In vortex merger, a PINN trained with a 128^2 Poisson grid resolves the 512^2 pseudo-spectral Fourier spectra until the 128^2 spectral simulation breaks down.The 128-resolution spectral run differs from the N = 512 reference by about 20% in relative L2 norm at t = 8, despite near-machine-precision conservation.Source: §4.2.2 · p. 16
4.3 Vlasov-Poisson
The Vlasov–Poisson tests show that learned flows reproduce the Landau-damping and two-stream dynamics, while the Eulerian PINN is less successful. Symplectic flows track the results more closely and conserve invariants more accurately than MLP flows.
Landau damping
- The SympNet flow is indistinguishable from the reference on Landau damping through T = 24; the MLP flow deviates slightly.Both learned flows follow the theoretical damping rate and reference.Source: §4.3 · p. 17, Figure 10 · p. 18
- For Landau damping, the Eulerian PINN follows only the first electric-energy oscillations and smooths filaments, while its f error is about eight times the MLP flow’s.SympNet energy drift is ∼10^-5, comparable to the reference and fifty times smaller than the MLP flow’s; the Eulerian PINN drifts on every invariant.Source: §4.3 · p. 18
Computational cost
- Training the six MLP submaps takes about 63 minutes, SympNet submaps 214 minutes, and the Eulerian PINN 28 minutes on an NVIDIA A100 GPU.Submap cost grows linearly with its index because evaluating later intervals composes earlier maps.Source: §4.3 · p. 18
Two-stream instability
- In the two-stream test, both learned flows follow electric-energy growth, saturation, and slow decay, while the Eulerian PINN reproduces neither the energy nor f.The SympNet flow follows the energy more closely than the MLP flow.Source: §4.3 · p. 18, Figure 11 · p. 19
- At T = 24 in the two-stream test, L2-norm drift is 2 × 10^-2 for MLP, 5 × 10^-4 for SympNet, and 2 × 10^-5 for the reference.The SympNet flow is about an order of magnitude more accurate than MLP on energy and momentum after saturation.Source: §4.3 · p. 19
4.4 An axisymmetric drift-kinetic Vlasov-Poisson model in a tokamak
The preliminary tokamak drift-kinetic test applies the Lagrangian PINN to a reduced axisymmetric model and evaluates the geodesic acoustic mode. The PINN tracks its oscillation over about 16 periods, with frequency close to GYSELA and damping consistent with a finer-grid semi-Lagrangian reference.
Model and quantity of interest
- The test is a reduced, single-species, axisymmetric electrostatic drift-kinetic model with adiabatic electrons in a large-aspect-ratio circular tokamak.The model’s quantity of interest is the normalized potential averaged over a magnetic surface at mid-radius; its damped oscillation settles to a positive residual level.Source: §4.4 · pp. 19, 21
Rosenbluth-Hinton test
- Over T = 40, about 16 periods, the PINN fit gives ωG = 2.54, within 0.5% of GYSELA, and γG = 0.099, 45% above GYSELA but agreeing with SL at 0.102.The PINN residual level is AR = 0.052, versus 0.061 for SL and the Rosenbluth-Hinton value 0.068.Source: §4.4 · p. 21, Figure 13 · p. 22
Radial boundary treatment
- Because characteristics cross the radial boundaries and inflow conditions are absent, points outside the domain are projected to the nearest boundary at submap times.A backward characteristic may remain on the boundary over some subintervals and re-enter the domain later; semi-Lagrangian codes apply the projection at every time step instead.Source: §4.4 · pp. 21–22
Density computation
- To compute the small density perturbation accurately, the method integrates energy variation along the characteristic instead of evaluating energy at its approximate endpoint.Characteristic approximation error is multiplied by ε in the energy variation; projections can also cause energy jumps that the calculation accounts for.Source: §4.4 · p. 22
5 Conclusion
The paper concludes that learning and composing flow maps can outperform Eulerian PINNs across the tested transport problems, while structure-preserving networks improve invariant accuracy. The authors also identify computational cost in low-dimensional settings as a limitation.
Conclusion
- The method learns composed flow submaps and recovers the solution by pulling back the exact initial condition; field-dependent velocities are handled by Picard iteration.The composition carries increasing solution complexity rather than requiring one network to represent it.Source: §5 · p. 23
- Where both were run, Lagrangian PINNs clearly outperform Eulerian PINNs, by more than an order of magnitude on advection, and reproduce the two-stream instability that Eulerian PINNs fail to reproduce.Source: §5 · p. 23
- At equal parameter budget, symplectic networks are about an order of magnitude more accurate than MLPs on Vlasov–Poisson and conserve all Lp norms up to quadrature error.Preliminary drift-kinetic results capture the geodesic acoustic mode frequency within 0.5%.Source: §5 · p. 23
- In the low-dimensional settings tested, the method is not cost-competitive with grid-based solvers, and each added submap costs more when velocity depends on the solution.The cost of each new submap grows linearly with the number of previous ones in that setting.Source: §5 · p. 23
Author Contribution Statement
The contribution statement assigns roles including methodology, investigation, implementation, software, validation, visualization, writing, revision, funding acquisition, and supervision.
Author contributions
- The listed contribution areas include methodology, investigation, implementation, software, validation, visualization, writing, revision, funding acquisition, and supervision.Source: Author Contribution Statement · p. 23
- E. Franck is credited with the initial idea, methodology, implementation, and supervision.Source: Author Contribution Statement · p. 23
- P. Krah is credited with investigation, implementation, software, validation, visualization, and writing the original draft.Source: Author Contribution Statement · p. 23