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Amortized Equation Discovery in Hybrid Dynamical Systems
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Proposes AMORE, an end-to-end framework that jointly infers discrete modes and counts with mode-specific sparse polynomial equations (plus AMORE-MIO for multi-object systems) to discover equations in hybrid dynamical systems.
Problem
Equation discovery in hybrid dynamical systems, where the number of modes and switching times are unknown; prior methods use a two-stage cluster-then-SINDy pipeline that does not share information across fragments of the same mode.
Approach
Replace the two-stage clustering-then-regression paradigm with a one-stage amortized model: latent mode and count variables are inferred by forward-backward recursions, and each mode's sparse basis-function coefficients are learned jointly from all segments assigned to that mode, optimized by maximizing observation likelihood with L1 regularization. Extended to multi-object systems with latent edg
Key results
- On Mass-spring Hopper segmentation, AMORE reaches NMI 0.928 and ARI 0.967 versus 0.426 and 0.383 for Hybrid-SINDy.
- On the SIR dataset, AMORE achieves forecasting NMAE of 0.088 / 0.113 for Susceptible/Infected versus 0.257 / 0.273 for the best baseline (SVI).
- On six non-hybrid systems, AMORE gives better log10 NRMSE than SVI on 4 of 6 systems, with the ODE particle task showing AMORE-MIO best segmentation (NMI 0.453).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Jointly learn mode categories and mode-specific equations end to end rather than use two-stage Hybrid-SINDy | AMORE · Mass-spring Hopper | NMI: 0.426 -> 0.928 | Helped |
| Jointly learn mode categories and mode-specific equations end to end rather than use two-stage Hybrid-SINDy | AMORE · Mass-spring Hopper | NMAE (Location/Velocity): 0.068 / 0.075 -> 0.008 / 0.039 | Helped |
| Jointly learn mode categories and mode-specific equations end to end rather than use two-stage Hybrid-SINDy | AMORE · SIR disease | Accuracy: 0.538 -> 0.731 | Helped |
| Jointly learn mode categories and mode-specific equations end to end rather than use two-stage Hybrid-SINDy | AMORE · SIR disease | NMAE (Susceptible/Infected): 0.257 / 0.273 -> 0.088 / 0.113 | Helped |
| Add interaction modeling for multi-object systems | AMORE-MIO vs AMORE · ODE-driven Particle | NMI: 0.418 -> 0.453 | Helped |
| Add interaction modeling for multi-object systems | AMORE-MIO vs AMORE · ODE-driven Particle | NMAE / NRMSE (one-step): 0.184 / 0.265 -> 0.146 / 0.217 | Helped |
| Add interaction modeling for multi-object systems | AMORE-MIO vs AMORE · Salsa-dancing | Accuracy: 0.565 -> 0.583 | Helped |
| Add interaction modeling for multi-object systems | AMORE-MIO vs AMORE · Salsa-dancing | NMAE / NRMSE (multi-step): 0.334 / 0.373 -> 0.301 / 0.352 | Helped |
| Change polynomial candidate basis order from 2 to 3 or 5 (ablation) | AMORE · Mass-spring Hopper | NMI: 0.934; 0.936; 0.933 | No clear effect |
| Change maximum predefined mode count from 3 to 5 or 10 (ablation) | AMORE · Mass-spring Hopper | NMI: 0.934; 0.932; 0.937 | No clear effect |
| Remove count variables (ablation) | AMORE · Mass-spring Hopper | false-positive mode-switching behaviors | Hurt |
| Use polynomial basis functions without cos(x), sin(x), and e^x (ablation) | AMORE · Synthetic two-mode nonlinear system | equation fidelity | Hurt |
Surprising findings
- On non-hybrid systems, AMORE is competitive with SVI, which is designed for them, and wins on only 4 of 6 systems.
- On the Salsa-dancing video dataset, AMORE (single-object) segmentation is slightly below GRASS, and only the multi-object variant edges it out.
Limitations
- Mode transitions and basis functions limited to polynomial candidate libraries in the main experiments; more complex dynamics only shown in a small synthetic test.
- Relies on an L1 relaxation of the L0 sparsity penalty rather than exact sparse selection.
- Approximates dynamics with a discrete-time Euler step y_t = y_{t-1} + dy * Delta_t, assuming dynamics change little between steps.
Experimental setup
Datasets: Mass-spring Hopper SIR disease dataset ODE-driven particle dataset Salsa-dancing dataset Coupled linear Cubic oscillator Lorenz'63 Hopf bifurcation Selkov glycolysis Duffing oscillator
Benchmarks: Mass-spring Hopper SIR disease dataset ODE-driven particle dataset Salsa-dancing dataset Lorenz'63 Duffing oscillator
Metrics: NMI ARI Accuracy F1 RER NMAE NRMSE
Baselines: Hybrid-SINDy SVI LLMTime TimeGPT GRASS
Builds on: SINDy Hybrid-SINDy REDSDS GRASS
Closest papers by meaning
- Automatically Learning Hybrid Digital Twins of Dynamical Systems (NeurIPS 2024 · similarity 0.86)
- Learning Hybrid Dynamics Models with Simulator-Informed Latent States (AAAI 2024 · similarity 0.86)
- Identifying latent state transitions in non-linear dynamical systems (ICLR 2025 · similarity 0.85)
- CausalDynamics: A large‐scale benchmark for structural discovery of dynamical causal models (NeurIPS 2025 · similarity 0.85)
- DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction (ICML 2025 · similarity 0.85)
- Learning system dynamics without forgetting (ICLR 2025 · similarity 0.85)
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