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Neural Context Flows for Meta-Learning of Dynamical Systems
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Proposes Neural Context Flows, a meta-learning Neural ODE framework that Taylor-expands the vector field around other environments' latent contexts so contexts modulate each other, with proximal alternating minimization and uncertainty from candidate trajectories.
Problem
Neural ODEs struggle to generalize to out-of-distribution environments whose dynamics depend on unobserved parameter changes, especially with limited data per environment.
Approach
Learn one environment-agnostic vector field conditioned on per-environment latent contexts, and Taylor-expand it around contexts from a context pool so information flows between environments and contexts self-modulate; a second-order JVP-based expansion gives NCF-t2 and a first-order variant gives NCF-t1.
Key results
- NCF-t2 achieves state-of-the-art OoD MSE on 5 of 6 problems (LV, GO, SM, BT, NS), while CoDA is best on GS.
- On Sel'kov Model OoD, NCF-t2 reaches 2.03e-3 MSE versus CoDA 8.28e-3 and CAVIA 859.1e-3.
- On Navier-Stokes OoD, NCF-t2 reaches 2.79e-3 MSE versus CoDA 7.08e-3, using 310955 parameters versus 309241 for CoDA.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NCF instead of OFA | SP · SP Adapt | MSE: 115000 $\pm$ 3200 -> 0.0356 $\pm$ 0.001 | Helped |
| Use NCF instead of OPE | SP · SP Adapt | MSE: 459.0 $\pm$ 345.0 -> 0.0356 $\pm$ 0.001 | Helped |
| Use NCF-t2 instead of CoDA | NCF-t2 · LV OoD | MSE: 2.19$\pm$0.78 -> 1.99$\pm$0.31 | Helped |
| Use NCF-t2 instead of CoDA | NCF-t2 · GO OoD | MSE: 4.22$\pm$4.21 -> 2.83$\pm$0.23 | Helped |
| Use NCF-t2 instead of CoDA | NCF-t2 · SM OoD | MSE: 8.28$\pm$0.29 -> 2.03$\pm$0.12 | Helped |
| Use NCF-t2 instead of CoDA | NCF-t2 · BT OoD | MSE: 19.47$\pm$11.6 -> 3.77$\pm$0.15 | Helped |
| Use NCF-t2 instead of CoDA | NCF-t2 · GS OoD | MSE: 0.75$\pm$0.65 -> 3.40$\pm$0.51 | Hurt |
| Use NCF-t2 instead of NCF-t1 (ablation) | NCF · SM OoD | MSE: 677.2$\pm$18.7 -> 2.03$\pm$0.12 | Helped |
| Use NCF-t2 instead of NCF-t1 (ablation) | NCF · GO OoD | MSE: 19.4$\pm$1.24 -> 2.83$\pm$0.23 | Helped |
| Use NCF-t2 instead of NCF-t1 (ablation) | NCF · LV OoD | MSE: 7.92$\pm$1.04 -> 1.99$\pm$0.31 | Helped |
| Use NCF-t2 instead of NCF-t1 (ablation) | NCF · NS OoD | MSE: 2.83$\pm$0.06 -> 2.79$\pm$0.09 | No clear effect |
Surprising findings
- NCF-t1 (first-order) fails on non-linear problems such as SM, performing much worse than NCF-t2, indicating second-order expansion matters.
- CoDA is best on Gray-Scott OoD, the only problem where NCF-t2 is not state-of-the-art.
Limitations
- Requires tuning many new hyperparameters such as context size, context pool size, and proximity coefficient.
- Assumes vector field is smooth in the context argument; fails for fundamentally discontinuous dynamics.
- Context pool cost scales quadratically (O(m^2)) when p=m.
Experimental setup
Datasets: Simple Pendulum Lotka-Volterra Glycolytic-Oscillator Sel'kov Model Brusselator Gray-Scott Navier-Stokes
Benchmarks: Simple Pendulum Lotka-Volterra Glycolytic-Oscillator Sel'kov Model Brusselator Gray-Scott Navier-Stokes
Metrics: MSE MAPE 3-sigma coverage
Baselines: CAVIA CoDA OFA OPE LEADS NCF-t1
Builds on: Neural ODE MAML CAVIA CoDA CAMEL
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