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ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence
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Proposes ControlSynth Neural ODEs (CSODEs), a NODE class with a sum of subnetworks plus a control term, gives LMI-based convergence conditions, and reports better prediction than NODE/ANODE/SONODE/MLP/RNN on Hindmarsh-Rose, Gray-Scott reaction-diffusion, and shallow water simulations.
Problem
Modeling highly nonlinear, scalable NODE dynamics with guaranteed convergence, and predicting physical dynamical systems (neuron, reaction-diffusion, shallow water PDEs) from time series.
Approach
Extends NODEs with a linear term plus a sum of activation subnetworks A_j f_j(W_j x) and an extra control term g(u(t)); convergence is certified via linear matrix inequalities derived from a Lyapunov/ISS-style argument under sector-type activation assumptions.
Key results
- On the Hindmarsh-Rose task, CSODE-Adapt reaches MSE 0.408 and R2 0.887 versus 0.739 MSE for the best non-CSODE NODE variant SONODE.
- On the Reaction-Diffusion task, CSODE-Adapt achieves MSE 5.635e-3 and CD 0.837, versus NODE MSE 1.134e-2 and CD 1.311.
- On MNIST, CSODE reaches 0.39% test error versus 0.42% for NODE with the same ~0.22M parameters; on CharacterTrajectories, Neural CDE outperforms CSODE (e.g., 97.8% vs 97.3% at 30% missing).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CSODE instead of SONODE | Hindmarsh-Rose prediction · Hindmarsh-Rose | MAE: 0.561 -> 0.470 | Helped |
| Use CSODE instead of SONODE | Reaction-Diffusion prediction · Reaction-Diffusion | MSE: 8.145e-3 -> 6.365e-3 | Helped |
| Use CSODE instead of SONODE | Shallow Water prediction · Shallow Water | MSE: 0.065 -> 0.041 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Hindmarsh-Rose | MSE: 0.783 -> 0.408 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Hindmarsh-Rose | MAE: 0.470 -> 0.370 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Hindmarsh-Rose | R²: 0.758 -> 0.887 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Reaction-Diffusion | MSE: 6.365e-3 -> 5.635e-3 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Reaction-Diffusion | MAE: 0.058 -> 0.035 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Reaction-Diffusion | CD: 0.939 -> 0.837 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Shallow Water | MSE: 0.041 -> 0.031 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Shallow Water | MAE: 0.16 -> 0.15 | Helped |
| Replace the control-term rate network with two convolutional layers (ablation) | CSODE · Shallow Water | CD: 2.27 -> 1.89 | Helped |
Surprising findings
- Neural CDE outperforms CSODE on irregular-observation tasks, which the authors acknowledge.
- The LMI-based convergence conditions are reported as verified numerically on trained models (>99% satisfiability on 500 samples), not analytically for the trained networks.
Limitations
- CSODEs are more sensitive to learning-rate choice due to complex architecture.
- Inductive biases of CSODEs vary by application and may not be preferable in all settings.
- Reported convergence guarantees rely on LMIs that are hard to verify analytically for trained models; empirical verification used a sampled subset.
Experimental setup
Datasets: MNIST CharacterTrajectories PhysioNet Sepsis
Benchmarks: Hindmarsh-Rose Reaction-Diffusion Shallow Water Equations MNIST CharacterTrajectories PhysioNet Sepsis Prediction
Metrics: MSE MAE R2 Chamfer Distance test error AUC test accuracy FLOPS time memory
Baselines: NODE ANODE SONODE MLP RNN TLODE Transformer Neural CDE ODE-RNN
Builds on: Neural ODE Augmented Neural ODE Second Order Neural ODE Neural CDE Latent ODE
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