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Parametric model reduction of mean-field and stochastic systems via higher-order action matching
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Learns parameter-dependent population dynamics as gradient fields via an action-matching-style variational loss with higher-order time quadrature (Simpson/Gauss-Legendre) and Monte Carlo over samples and parameters, enabling fast trajectory sampling across physics parameters.
Problem
Surrogate modeling of population dynamics of stochastic/mean-field physical systems that depend on physics parameters; existing conditioning-based diffusion/flow models need separate inference per time step and parameter, and action matching with Monte Carlo time integration is unstable.
Approach
Extends action matching to parameter-dependent population dynamics by learning a parametric gradient vector field s(t,mu) via an energy (optionally entropic/Schrödinger-bridge) loss, with the time integral estimated by higher-order quadrature (composite Simpson, Gauss-Legendre) and CoLoRA weight modulation over time and parameters, so trajectories are generated by one SDE integration.
Key results
- HOAM-S achieves electric-energy relative error 0.078 on two-stream vs 0.275 for AM and 1.44 for CFM, with 6 s inference runtime vs 139 s for CFM and 1142 s for NCSM.
- On the 9D chaotic system, HOAM-S reaches Sinkhorn 0.214 vs 80.1 for AM, while AM fails to converge on the 6D strong Landau damping problem.
- HOAM gives about 2 orders of magnitude inference speedup over the full six-dimensional particle-in-cell model on strong Landau damping (runtime rises from 6 to 8 s).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use higher-order quadrature (HOAM-S) instead of Monte Carlo time integration (AM) (ablation) | Action matching · Two-stream instability | e.e.: 0.275 -> 0.078 | Helped |
| Use higher-order quadrature (HOAM-S) instead of Monte Carlo time integration (AM) (ablation) | Action matching · Bump-on-tail instability | e.e.: 0.892 -> 0.427 | Helped |
| Use higher-order quadrature (HOAM-G) instead of Monte Carlo time integration (AM) (ablation) | Action matching · Strong Landau damping | e.e. | Helped |
| Use higher-order quadrature (HOAM-S) instead of Monte Carlo time integration (AM) (ablation) | Action matching · 9D chaos | sinkhorn: 80.1 -> 0.214 | Helped |
| Use Simpson's quadrature rather than Gauss-Legendre quadrature (ablation) | HOAM · Two-stream instability | e.e.: 0.208 -> 0.078 | Helped |
| Use Simpson's quadrature rather than Gauss-Legendre quadrature (ablation) | HOAM · Strong Landau damping | e.e.: 0.447 -> 0.641 | Hurt |
| Use higher-order quadrature for the training time integral (ablation) | Action matching · Harmonic oscillator | training stability | Helped |
| Use CoLoRA modulation instead of FiLM or a concatenation MLP (ablation) | Vector-field parameterization · Harmonic oscillator | mean Wasserstein distance | Helped |
Surprising findings
- Monte Carlo time integration in action matching (AM) causes training divergence for some seeds, while higher-order quadrature gives stable loss curves.
- Conditional flow matching and NCSM provide no speedup over full numerical models in these problems.
Limitations
- Quadrature needs samples at specific time points, so sparse time sampling (e.g. computational biology data) limits accuracy; interpolation is required otherwise.
- Only minimal kinetic energy (or entropic variant) vector fields are considered; other energy choices are not explored.
Experimental setup
Datasets: Vlasov-Poisson particle-in-cell simulations (two-stream, bump-on-tail, strong Landau damping via Struphy) Rayleigh-Bénard nine-dimensional chaotic system samples harmonic oscillator simulations aharmonic trap particle SDE samples
Benchmarks: two-stream instability bump-on-tail instability strong Landau damping 9D chaos harmonic oscillator aharmonic trap particles
Metrics: relative error in electric energy inference runtime (s) Sinkhorn divergence mean Wasserstein distance
Baselines: action matching (AM) noise-conditioned score matching (NCSM) conditional flow matching (CFM) FiLM
Builds on: action matching Benamou-Brenier formula CoLoRA conditional flow matching noise-conditioned score matching
Closest papers by meaning
- Wasserstein residuals: Learning Gradient Flows from Population Dynamics (NeurIPS 2026 · similarity 0.87)
- First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems (NeurIPS 2026 · similarity 0.87)
- Sampling in Unit Time with Kernel Fisher-Rao Flow (ICML 2024 · similarity 0.87)
- Curly Flow Matching for Learning Non-gradient Field Dynamics (NeurIPS 2025 · similarity 0.87)
- Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers (NeurIPS 2025 · similarity 0.86)
- Generating Physical Dynamics under Priors (ICLR 2025 · similarity 0.86)
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