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Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching
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Proposes APPS, which actively selects informative initial-condition regions via phase portrait sketching of candidate ODEs and uses a Transformer decoder trained with REINFORCE to discover ODEs from queried trajectories.
Problem
Symbolic discovery of ODEs from trajectory data typically relies on fixed pre-collected datasets, causing overfitting; active querying is hard because chaotic dynamics (butterfly effect) require dense sampling of initial conditions.
Approach
Instead of selecting individual initial conditions, select an informative region in phase space by sketching phase portraits of candidate ODEs (region where predicted trajectories diverge most), then sample a batch of initial conditions from that region to query a data oracle, and use the queried data as the REINFORCE reward for a sequential grammar-based decoder.
Key results
- On noiseless Strogatz n=1 APPS median NMSE 2.06E-6 vs SINDy 1.90E-4 and ODEFormer 0.0303; APPS achieves lowest median NMSE on most noiseless settings.
- On noisy Strogatz (sigma^2=0.01) APPS n=1 median NMSE 7.75E-4 vs ODEFormer 0.139; on irregular Strogatz n=1 1.06E-6 vs ODEFormer 0.031.
- In the active-strategy comparison, APPS achieves ranking distance 0.08 with 5.2 s running time and 3.76 MB peak memory, versus QbC 0.13 / 13.4 s / 51 MB and CoreSet 0.22 / 4.3 s / 2.74 GB.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use APPS rather than passive baselines | ODE discovery · Strogatz and ODEBase, noiseless settings | median NMSE | Mixed |
| Use APPS rather than passive baselines | ODE discovery · Strogatz, noisy settings | median NMSE | Helped |
| Use APPS rather than passive baselines | ODE discovery · Strogatz, irregular-time settings | median NMSE | Mixed |
| Use phase-portrait region selection rather than QbC | Active data query strategy · Active-strategy benchmark | ranking-based distance: 0.13 -> 0.08 | Helped |
| Use phase-portrait region selection rather than CoreSet | Active data query strategy · Active-strategy benchmark | running time: 4.3 sec -> 5.2 sec | Hurt |
| Use phase-portrait region selection rather than QbC | Active data query strategy · Active-strategy benchmark | peak memory: 51 MB -> 3.76 MB | Helped |
Surprising findings
- SINDy achieves lower median NMSE than APPS on the n=2 Strogatz noiseless setting (0.217 vs 0.2912), so APPS is not best on every setting.
- CoreSet is faster than APPS in running time (4.3 s vs 5.2 s) despite using far more memory (2.74 GB).
Limitations
- Phase portrait sketching may suffer resolution issues: fine dynamics can be missed if region width is too large or too few points are sampled.
- Unclear whether the phase portrait sketching idea applies to partial differential equations.
- Assumes access to a data oracle that can be queried with chosen initial conditions, which may be expensive in real experiments.
Experimental setup
Datasets: Strogatz dataset ODEBase dataset
Benchmarks: Strogatz ODEBase
Metrics: median NMSE NMSE quantiles Kendall tau ranking distance running time peak memory
Baselines: SINDy ODEFormer SPL ProGED E2ETransformer QbC CoreSet
Builds on: ODEFormer SPL ProGED SINDy REINFORCE
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