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Learning Stochastic Multiscale Models
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Learns stochastic multiscale latent SDE models (macroscale coarse state plus microscale latent state) from noisy observations via amortized simulator-free variational inference with a product-of-experts likelihood, outperforming coarse DNS, DMD, POD-SINDy and implicit-scale models on four PDE test cases.
Problem
Predicting multiscale PDE dynamics that require resolving a wide range of length and time scales, where direct simulation at the finest scale is computationally infeasible and coarse models lose sub-grid effects.
Approach
Decompose the latent state into a coarse-grid macroscale state and a learned microscale latent state evolving under coupled Itô SDEs, with a product-of-experts likelihood that forces the microscale state to explain only the residual unexplained by the macroscale, trained by simulator-free amortized SVI with a reparametrized ELBO.</218core_idea>
Key results
- On the advecting wave, the n_eta=5 multiscale model reduces mean error from 0.469 (best baseline POD-SINDy) to 0.079 (83% reduction).
- On KdV, n_eta=5 reduces mean error from 0.324 (implicit scale) to 0.042 (87% reduction).
- On the shallow water equations, n_eta=5 reduces mean error from 0.426 (implicit scale) to 0.152 (64% reduction); on cylinder flow it gives 0.009 vs DMD 0.016 (44% lower).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the multiscale model with n_eta=5 instead of POD-SINDy | Wave 1D prediction · Wave 1D | normalized error: 0.469 to 0.079 | Helped |
| Use the multiscale model with n_eta=5 instead of the implicit-scale model | KdV 1D prediction · KdV 1D | normalized error: 0.324 to 0.042 | Helped |
| Use the multiscale model with n_eta=5 instead of DMD | Cylinder 2D prediction · Cylinder 2D | normalized error: 0.016 to 0.009 | Helped |
| Use the multiscale model with n_eta=5 instead of the implicit-scale model | SWE 2D prediction · SWE 2D | normalized error: 0.426 to 0.152 | Helped |
| Increase n_eta from 1 to 2 (ablation) | Our approach · Wave 1D | normalized error: 0.517 -> 0.210 | Helped |
| Increase n_eta from 1 to 2 (ablation) | Our approach · KdV 1D | normalized error: 0.197 -> 0.077 | Helped |
| Increase n_eta from 1 to 2 (ablation) | Our approach · Cylinder 2D | normalized error: 0.038 -> 0.015 | Helped |
| Increase n_eta from 1 to 2 (ablation) | Our approach · SWE 2D | normalized error: 0.543 -> 0.202 | Helped |
| Increase n_eta from 2 to 3 (ablation) | Our approach · Wave 1D | normalized error: 0.210 -> 0.105 | Helped |
| Increase n_eta from 2 to 3 (ablation) | Our approach · KdV 1D | normalized error: 0.077 -> 0.059 | Helped |
| Increase n_eta from 2 to 3 (ablation) | Our approach · Cylinder 2D | normalized error: 0.015 -> 0.011 | Helped |
| Increase n_eta from 2 to 3 (ablation) | Our approach · SWE 2D | normalized error: 0.202 -> 0.190 | Helped |
Surprising findings
- DMD achieves competitive performance on the periodic cylinder flow (error 0.016), close to the multiscale model.
- The implicit-scale model (n_eta=0) sometimes underperforms the best reduced-order baselines, e.g., on the 1D advecting wave and cylinder flow.
Limitations
- Microscale state is a global compression, which may be suboptimal for problems with local sub-grid dynamics such as 3D turbulence.
- Extension to chaotic systems and long-term stability is not addressed.
- Markovian latent SDE is an approximation of non-Markovian coarse-grained dynamics.
Experimental setup
Datasets: Simulated 1D advection wave Simulated 1D KdV equation Von Karman vortex street (Günther et al.) PDEBench shallow water equations
Benchmarks: Advecting wave 1D KdV 1D Cylinder 2D SWE 2D
Metrics: normalized error mean and std error vs. computational cost (NFE x state dimension)
Baselines: Coarse DNS DMD POD-SINDy Implicit scale model
Builds on: Course and Nair amortized SVI for SDEs Mori-Zwanzig formalism SINDy DMD POD
Closest papers by meaning
- Closing the Approximation Gap in Simulation-free Latent SDEs (NeurIPS 2026 · similarity 0.89)
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