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No Equations Needed: Learning System Dynamics Without Relying on Closed-Form ODEs
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Proposes direct semantic modeling, implemented as Semantic ODE, which predicts the shape-based semantic representation (motif compositions and properties) of a 1D dynamical system directly from initial conditions instead of discovering a closed-form ODE and analyzing it.
Problem
Closed-form ODE discovery requires costly post-hoc analysis to understand system behavior, and the link between equation form and behavior makes editing and incorporating behavioral constraints hard.
Approach
Predict the semantic representation (a composition of dynamical motifs plus shape properties) as a function of the initial condition with a semantic predictor, then map it to a C2 trajectory with a deterministic trajectory predictor (cubic splines), so behavior can be inspected, constrained and edited directly.
Key results
- On the pharmacokinetic dataset, Semantic ODE achieves in-domain RMSE 0.015 and the lowest RMSE on 4 of 5 synthetic benchmarks under high and low noise, outperforming compact SINDy/WSINDy variants.
- After editing the property map to force the horizontal asymptote to 0, Semantic ODE's out-domain error drops to 0.015, comparable to its in-domain error.
- On the Duffing oscillator, Semantic ODE achieves RMSE 0.096 (low noise) vs 0.176 for the best baseline (Neural Laplace).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Semantic ODE instead of Neural ODE | Semantic ODE · General ODE, low noise | RMSE: 0.029 -> 0.016 | Helped |
| use Semantic ODE instead of Neural ODE | Semantic ODE · General ODE, high noise | RMSE: 0.075 -> 0.068 | Helped |
| use Semantic ODE instead of Neural Laplace | Semantic ODE · Mackey-Glass (DDE), low noise | RMSE: 0.057 -> 0.037 | Helped |
| use Semantic ODE instead of Neural Laplace | Semantic ODE · Mackey-Glass (DDE), high noise | RMSE: 0.094 -> 0.077 | Helped |
| use Semantic ODE instead of WSINDy-5 | Semantic ODE · Logistic Growth, low noise | RMSE: 0.010 -> 0.015 | Hurt |
| use Semantic ODE instead of Neural ODE | Semantic ODE · Logistic Growth, high noise | RMSE: 0.197 -> 0.198 | No clear effect |
| add semantic composition bias $(s_{-+c},s_{++u})$ (ablation) | Semantic ODE · Tumor growth (real) | RMSE: 0.234 -> 0.229 | Helped |
| add semantic composition bias $(s_{+-c},s_{--c},s_{-+h})$ (ablation) | Semantic ODE · Drug concentration (real) | RMSE: 0.264 -> 0.243 | Helped |
| fix the horizontal asymptote at 0 (ablation) | Semantic ODE · Pharmacokinetic dataset, out-domain ($t>1$) | RMSE: 0.033 -> 0.015 | Helped |
| fix the horizontal asymptote at 0 (ablation) | Semantic ODE · Pharmacokinetic dataset, in-domain ($t\leq 1$) | RMSE: 0.016 -> 0.018 | Hurt |
Surprising findings
- Compact SINDy equations (e.g., 5 terms) perform poorly on the pharmacokinetic data, and the best SINDy variant needs up to 15 terms, which overfits (out-domain error 77.6).
Limitations
- Limited to one-dimensional systems; multidimensional extension only shown as proof of concept.
- Cannot represent oscillatory behavior such as sin with a finite composition, so chaotic systems like the Duffing oscillator are only modeled on bounded domains.
- Semantic representation requires choosing motifs, composition library, and property parametrizations manually.
Experimental setup
Datasets: logistic growth simulated data pharmacokinetic dataset Mackey-Glass DDE integro-DE Duffing oscillator tumor growth dataset drug concentration dataset SIR epidemiological model simulations
Benchmarks: logistic growth general ODE pharmacokinetic model Mackey-Glass (DDE) integro-DE Duffing oscillator tumor growth (real) drug concentration (real)
Metrics: RMSE
Baselines: SINDy WSINDy PySR Neural ODE Neural Laplace DeepONet
Builds on: dynamical motifs framework SINDy PySR
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