Structure-Informed Data-Driven Reduced-Order Modeling of Scalar Hyperbolic Conservation Laws via Kinetic Defect Measure

Marissa Llamas; Jan Fuhg; Hannah Lu

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Structure-Informed Data-Driven Reduced-Order Modeling of Scalar Hyperbolic Conservation Laws via Kinetic Defect Measure

Marissa Llamas, Jan Fuhg, Hannah Lu

arXiv:2609.16525v1Sep 15, 2026math.NAphysics.comp-ph

TL;DR

Transport-dominated shocks are poorly represented by low-dimensional linear subspaces, motivating a structure-informed ROM for scalar hyperbolic conservation laws. The method prescribes known characteristic transport, learns shock geometry and registered defect dynamics, and accurately predicts shock evolution beyond training while preserving mass and entropy behavior.

Problem
Moving fronts and shocks are poorly represented by low-dimensional linear subspaces, making reduced-order modeling of transport-dominated systems challenging.
Method
The method removes known characteristic transport, extracts and registers localized defect-driven dynamics in shock-attached coordinates, and learns separate ROMs for shock geometry and defect evolution.
Results
The ROM accurately reconstructs and predicts shock evolution beyond the training interval while maintaining close agreement with reference mass and quadratic entropy.
Takeaways & Limitations
Kinetic-defect structure restricts model reduction to the localized departure from free transport while retaining accurate nonlinear shock dynamics.
Takeaways & Limitations
The current framework focuses on solutions whose active shock manifold can be represented within a single graph chart; multiple shocks, rarefactions, and topology changes require extensions.

Abstract

from arXiv · show

Reduced-order modeling of transport-dominated systems remains challenging because moving fronts and shocks are poorly represented by low-dimensional linear subspaces. We develop a structure-informed data-driven reduced-order model(ROM) for scalar hyperbolic conservation laws based on the kinetic defect formulation. This formulation separates the nonlinear dynamics into known characteristic transport and a kinetic entropy defect localized on the shock manifold. We exploit this structure by first removing the known transport from the solution snapshots. We then extract and register the remaining defect-driven dynamics in a shock-attached coordinate system. Separate ROMs are used to evolve the shock geometry and the registered defect-driven source. During prediction, the predicted shock geometry is used to inverse-register the learned defect-driven source, which advances the kinetic state and recovers the physical solution. Numerical examples in one and two spatial dimensions demonstrate accurate reconstruction and prediction of nonlinear transport with shocks, including evolution beyond the training interval, while accurately capturing the mass and entropy-dissipation behavior of the reference solution.

1. Introduction

Transport-dominated solutions with moving fronts and shocks are difficult to compress using linear ROMs. The paper addresses this by combining kinetic defect structure, characteristic transport, and shock-attached registration.

Moving fronts and shocks yield slowly decaying Kolmogorov n-widths, so simple translating structures can require many spatial modes.
Existing remedies include adaptive bases, alignment, registration, nonlinear coordinate transformations, and nonlinear-manifold ROMs, but each changes the representation used for reduction.
Hodograph-based DMD incorporates shock information but relies on invertibility or monotone solution branches and does not extend straightforwardly to multiple spatial dimensions.
The kinetic formulation lifts the conservation law to linear transport with known characteristic velocity, while a kinetic entropy defect represents shock-associated entropy production and exact physical recovery follows by kinetic integration.
The defect is nonnegative, kinetically bounded, vanishes in smooth regions, and is spatially supported on the lower-dimensional shock manifold for piecewise smooth solutions.
The proposed ROM removes known characteristic transport, registers localized defect dynamics in shock-attached coordinates, and learns separate models for shock geometry and defect-driven evolution.

2. Kinetic formulation of scalar hyperbolic conservation laws

The kinetic formulation rewrites scalar conservation laws as known transport plus an entropy-constrained defect. The defect is localized on shocks, enabling characteristic pullback and shock-supported reduced representations.

Kinetic formulation of scalar hyperbolic conservation laws: Scalar conservation laws evolve a conserved quantity through a flux, while nonlinear characteristic intersection can create shocks and require an entropy condition to select the physical weak solution.
2.1. Kinetic entropy defect: The kinetic lift introduces an auxiliary coordinate and represents the solution through a kinetic function whose physical state is recovered exactly by integrating over that coordinate.
2.1. Kinetic entropy defect: The lifted equation separates transport with velocity f′(ξ), determined by the flux, from a kinetic defect measure that supplies the entropy-producing correction.
2.1. Kinetic entropy defect: The kinetic defect is nonnegative and constrained by entropy admissibility, so one measure represents dissipation for every convex entropy pair rather than serving as an arbitrary closure.
2.2. Characteristic representation: Along free characteristics, the pullback P remains constant where the defect vanishes, and any temporal variation of P measures the defect-driven departure from free transport.
2.2. Characteristic representation: The characteristic decomposition consists of transported initial data plus a correction accumulated from the kinetic defect along each characteristic.
2.3. Shock-supported structure of the KED: For sufficiently regular shocks, the defect measure is supported on the moving shock manifold, with one-dimensional shocks represented by trajectories and multiple shocks by summed contributions between topology changes.
2.3. Shock-supported structure of the KED: The ROM uses this localization as a reduction principle, learning shock geometry and registered source dynamics from characteristic-pullback changes instead of explicitly recovering the defect measure.

3. Data-driven ROMs via KED

The ROM prescribes known kinetic transport analytically and learns only the localized defect-driven dynamics. It registers the source around the evolving shock manifold before reducing its dynamics.

The framework lifts snapshots to kinetic space, pulls them back along free characteristics, estimates the defect source from temporal changes, and registers its moving support in shock-attached coordinates.
Separate reduced models learn the shock geometry and registered defect-driven source, while the analytically known characteristic transport is not learned from data.

3.1. Regularized kinetic lifting and decoding

The method uses a regularized kinetic lift for nonnegative solution snapshots and decodes the predicted kinetic state by quadrature. Finite-resolution decoding bias is corrected through monotone interpolation.

For nonnegative snapshots, the kinetic function becomes a subgraph indicator over the interval from zero to the physical state.
The lift explicitly bounds the occupied kinetic interval using Heaviside factors, with one factor enforcing positive kinetic levels and the other limiting levels below u_n.
A smooth approximation Hε replaces the discontinuous Heaviside function to produce the diffuse kinetic lift used numerically.
The kinetic grid uses discrete levels and quadrature weights, with a slightly enlarged kinetic domain resolving diffuse tails near the interval endpoints.
The physical state is decoded by integrating the kinetic representation, but finite kinetic resolution introduces a deterministic bias corrected by precomputed monotone interpolation.
Figure 1 shows selected diffuse kinetic slices alongside the physical state recovered from kinetic-coordinate integration at t = 0.24.

3.2. Characteristic pullback and empirical defect-driven evolution

The characteristic pullback removes known kinetic transport, so its temporal variation provides an empirical measure of defect-driven evolution. The method evaluates this variation directly at midpoint locations and separates pre-event transport from post-breaking defect dynamics.

Characteristic variation along known kinetic paths measures the departure from free transport generated by the kinetic entropy defect.
The empirical defect-driven evolution is approximated over each time interval at the temporal midpoint.
The characteristic difference is evaluated directly at desired midpoint locations rather than by constructing a complete characteristic-coordinate pullback grid.
Before shock formation, the exact pullback is stationary and the diffuse characteristic difference remains small, so reduced shock and defect models use post-breaking midpoint snapshots.
Pre-event evolution is retained in reconstruction but governed by homogeneous characteristic transport instead of nearly zero defect data.

3.3. Shock-manifold extraction and registration

The method registers defect activity attached to an evolving shock by extracting its geometry and expressing the source in shock-attached coordinates. This separates shock motion and deformation from localized defect-shape changes for reduced modeling.

Registration separates shock translation, front deformation, and localized defect-shape changes by representing the defect-driven field in a common shock-attached coordinate system.
The construction produces a physical shock embedding and a registered defect-driven source using tangential coordinate θ and signed shock-normal coordinate η.
Defect-based shock-manifold embedding: Active kinetic levels are aggregated with quadrature weights to form a scalar event-density indicator for shock-location extraction.
Defect-based shock-manifold embedding: Within a local graph chart, the strongest defect response identifies the raw transverse shock position, which is refined and spatially fitted into the shock representation.
Defect-based shock-manifold embedding: Tangential registration maps variable shock extents to a fixed reference domain, while the embedding records physical location and tangential extent.
Registered defect-driven source: The registered source is evaluated in shock-attached coordinates through X(θ,η)=c(θ)+ηd, avoiding construction of the complete physical-kinetic defect field.
ROMs for the shock geometry and registered defect-driven source: Two independent DMD models reduce the shock embedding and registered source; in one dimension, the shock location is instead fitted polynomially.
Inverse registration: During prediction, the registered source is inverse-registered to physical midpoint coordinates and inserted into the characteristic kinetic update, with reconstructed defect set to zero outside the chart.

3.4. Computational workflow

Algorithm 1 builds the ROM offline from kinetic lifts, characteristic defect sources, shock extraction, registration, and independent reduced models. Online, it evolves those models, inverse-registers the source, advances the kinetic field, and decodes the physical solution.

Offline construction lifts solution snapshots, evaluates the empirical defect source, extracts and registers the shock manifold, and builds independent ROMs for geometry and registered source.
Algorithm 1 is identified as the kinetic-defect reduced-order model and outputs physical reconstructions together with the relevant grids.
Online prediction evolves the shock-geometry and registered-defect models, inverse-registers the predicted source, advances the kinetic field, and decodes the physical solution.

4. Numerical examples

Four one- and two-dimensional Burgers and Buckley–Leverett examples test shock formation, propagation, and multidimensional front evolution, including prediction beyond finite post-breaking training intervals. Across the examples, the ROM captures shock geometry and defect-driven dynamics while maintaining controlled solution errors and close global mass and quadratic-entropy behavior.

4. Numerical examples: The study uses four increasing-complexity problems, with finite post-breaking training windows followed by prediction intervals to assess extrapolative capability.The examples cover Burgers and Buckley–Leverett fluxes in one and two spatial dimensions.
4.1. Ramp–Riemann initial condition: In the ramp–Riemann Burgers example, relative L2 error remains small while mass and quadratic entropy closely follow analytic references throughout prediction.The solution reproduces pre-breaking characteristic evolution and tracks the post-breaking shock beyond the training interval.
4.1–4.2. One-dimensional Burgers examples: The ROM accurately predicts shock formation and propagation beyond training in one-dimensional Burgers examples, including rarefaction and time-varying shock speed without spurious oscillations.Errors remain localized near the discontinuity because small position errors create localized discrepancies.
4.3. Two-dimensional Burgers equation with Gaussian initial data: The two-dimensional Burgers ROM captures formation and propagation of a curved shock front beyond training, with controlled error and mass and quadratic entropy close to the reference.The result demonstrates prediction of both evolving shock geometry and associated defect-driven dynamics.
4.4. Two-dimensional Buckley–Leverett equation: The two-dimensional Buckley–Leverett ROM reproduces transverse deformation and propagation of the saturation front, maintaining strong agreement throughout prediction.Independent reduced models use ranks rc = 2 and rG = 43 under the 99% snapshot-energy criterion.
4. Numerical examples: Across the examples, total mass and quadratic entropy remain close to reference behavior, preserving principal global balance properties during heterogeneous front prediction.These diagnostics complement relative L2 solution error by testing conservative and entropy-dissipative behavior.

5. Conclusions

The paper develops a kinetic-defect ROM that separates known characteristic transport from shock-associated defect dynamics, enabling accurate prediction beyond training while preserving mass and entropy dissipation. Its current scope is limited to active shock manifolds representable in a single graph chart, motivating multi-chart extensions for interacting waves.

The kinetic-defect ROM accurately reconstructs and predicts shock-dominated solutions beyond the training interval while retaining the reference solutions’ global mass and entropy-dissipation behavior.The framework uses prescribed characteristic transport, separate ROMs for shock geometry and registered defect dynamics, and shock-attached coordinates.
The current framework focuses on solutions whose active shock manifold can be represented within a single graph chart.
Extending the method to multiple shocks, rarefactions, and wave interactions requires multiple local graph charts plus mechanisms to identify, evolve, couple, create, and remove charts adaptively.