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Learning Macroscopic Dynamics from Partial Microscopic Observations
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Learns macroscopic (latent) closure dynamics from partial microscopic force computations by mapping the macroscopic loss back to microscopic coordinates and using randomly sampled force subsets as unbiased stochastic estimates.
Problem
Learning macroscopic observable dynamics requires forces on all microscopic coordinates, which is computationally prohibitive for large systems (e.g., DFT-based molecular simulation, large PDE discretizations).
Approach
Map the latent macroscopic loss back to microscopic coordinates via the Moore-Penrose inverse of the encoder Jacobian, so that random subsets of microscopic forces (scaled by 1/p) give an unbiased stochastic estimate of the full loss; a condition-number penalty keeps the weighting well-behaved.
Key results
- On Predator-Prey, L_x with partial forces p=1/5 reaches test error ~1.34e-3 at 3e3 data vs 3.19e-3 for full-force L_z, and the paper reports partial-force training needs less data to reach error tolerance 3e-3.
- On Allen-Cahn (N=40000, p=1/25) the partial-force loss gives 3.98e-3 error versus 6.93e-3 for L_z.
- On Lennard-Jones (small, N=4800, p=1/16) partial-force loss gives 1.17e-3 vs 4.45e-3 for L_z; a large LJ system with 51200 atoms (N=307200, p=1/1024) reaches 4.96e-3 error using only 5000 partial-force configurations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| train with partial forces (p=1/5) instead of full forces | Macroscopic dynamics model · Predator-Prey | mean relative error: 3.19 ±0.60 × 10^{-3} -> 1.34 ±0.16 × 10^{-3} | Helped |
| train with partial forces (p=1/25) instead of full forces | Macroscopic dynamics model · Allen-Cahn | mean relative error: 6.93 ±2.80 × 10^{-3} -> 3.98 ±1.58 × 10^{-3} | Helped |
| train with partial forces (p=1/16) instead of full forces | Macroscopic dynamics model · Lennard-Jones (small) | mean relative error: 4.45 ± 2.03 × 10^{-3} -> 1.17 ± 0.18 × 10^{-3} | Helped |
| use partial forces (p=1/1024) instead of full-force training | Macroscopic dynamics model · Lennard-Jones (large) | mean relative error | Mixed |
| reduce the partial-label fraction from p=3/4 to p=1/4 (ablation) | Partial-force Predator-Prey model · Predator-Prey (6.0 ×10^3 training data) | mean relative error: 1.07 ± 0.18 × 10^{-3} -> 9.08 ± 0.25 × 10^{-4} | Helped |
| reduce the partial-label fraction from p=3/4 to p=1/2 (ablation) | Partial-force Predator-Prey model · Predator-Prey (6.0 ×10^3 training data) | mean relative error: 1.07 ± 0.18 × 10^{-3} -> 1.46 ± 0.59 × 10^{-3} | Hurt |
| reduce the partial-label fraction from p=1/2 to p=1/4 (ablation) | Partial-force Predator-Prey model · Predator-Prey (6.0 ×10^3 training data) | mean relative error: 1.46 ± 0.59 × 10^{-3} -> 9.08 ± 0.25 × 10^{-4} | Helped |
| reduce the partial-label fraction from p=1/4 to p=1/5 (ablation) | Partial-force Predator-Prey model · Predator-Prey (6.0 ×10^3 training data) | mean relative error: 9.08 ± 0.25 × 10^{-4} -> 1.13 ± 0.31 × 10^{-3} | Hurt |
| train using partial-force loss rather than macroscopic-coordinate loss | Macroscopic dynamics models with MLP, OnsagerNet, or GFINNs latent dynamics · Lennard-Jones (800 atoms) | test error | Helped |
| decrease p from 3/4 to 1/4 or 1/5 (ablation) | Partial-force Predator-Prey model · Predator-Prey | training data required to reach error tolerance | Helped |
Surprising findings
- Partial-force training with smaller p (1/4, 1/5) reached target error with less data than full-force training, suggesting full forces contain redundant information and partial forces let more configurations be explored.
- For larger Lennard-Jones systems, the number of force computations needed to reach the error tolerance did not grow as fast as system size, possibly because dynamics become less fluctuating.
Limitations
- Relies on a sparsity assumption; systems with global interactions (e.g., McKean-Vlasov) do not benefit.
- Encoder is an MLP rather than permutation-invariant, which is suboptimal for particle systems.
- Assumes a known training distribution D of microscopic configurations and deterministic microscopic dynamics.
Experimental setup
Datasets: Predator-Prey simulations (1D, N=100) Allen-Cahn simulations (N=40000) Lennard-Jones MD simulations via LAMMPS (800 to 51200 atoms)
Benchmarks: Predator-Prey system Allen-Cahn system Lennard-Jones system (small, 800 atoms) Lennard-Jones system (large, 51200 atoms)
Metrics: mean relative error of macroscopic observables number of force computations to reach error tolerance 3e-3
Baselines: full-force latent dynamics loss L_z OnsagerNet GFINNs MLP latent model
Builds on: OnsagerNet GFINNs LAMMPS
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