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Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations
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Proposes Balanced Neural ODEs (B-NODEs), a VAE-based latent Neural ODE that propagates variational parameters through time with a non-hierarchical prior, enabling adaptive nonlinear model order reduction and linear Koopman operator approximation for systems with time-varying inputs.
Problem
Building fast, low-order surrogate models of simulation systems with time-varying inputs; standard Latent ODEs fail to achieve state reduction when inputs are present, and Koopman approximations need predefined observable/latent dimensionality.
Approach
Combine a VAE information bottleneck (beta-weighted KL with a fixed Gaussian prior) with a Neural ODE that continuously propagates the mean and variance of the latent state through time, so the latent dimensionality emerges from beta rather than being predefined; linear ODE dynamics yield Koopman-style approximations.
Key results
- On the discretized heat flow example, B-NODE with 4 latent states reaches RMSE 0.3% versus 14.2% for Truncated Balanced Realization.
- On the thermal power plant, B-NODE (beta=0.05, constant variance) achieves the lowest reconstruction error of the compared models using 17 states instead of 102.
- B-NODE gives a 32x speed-up on CPU and 103x on GPU for batch simulation over the power plant model, and needs about 10x fewer ODE function evaluations than SS-NODE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use B-NODE β=0.01 instead of SS-NODE | Nonlinear surrogates · Power Plant | RMSE: 0.06 -> 0.02 | Helped |
| Use B-NODE β=0.01 instead of SS-NODE | Nonlinear surrogates · MuJoCo | RMSE: 0.43 -> 0.28 | Helped |
| Use B-NODE β=0.01 instead of Latent NODE β=0.1 | Nonlinear surrogates · Power Plant | state dimension: 128 -> 11 | Helped |
| Set β=0.0 instead of β=0.01 (ablation) | B-NODE · Power Plant | RMSE: 0.02 -> 18.2 | Hurt |
| Set β=0.0 instead of β=0.01 (ablation) | B-NODE · SHF | RMSE: 0.02 -> 0.01 | Helped |
| Set β=0.0 instead of β=0.01 (ablation) | B-NODE · Power Plant | state dimension: 11 -> 128 | Hurt |
| Increase β from 0.01 to 0.1 (ablation) | B-NODE · SHF | RMSE: 0.02 -> 0.04 | Hurt |
| Increase β from 0.01 to 0.1 (ablation) | B-NODE · Water Hammer | state dimension: 8 -> 4 | Helped |
| Increase β from 0.01 to 0.1 (ablation) | Linear B-NODE · Power Plant | RMSE: 0.20 -> 0.20 | No clear effect |
| Increase β from 0.01 to 0.1 (ablation) | Linear B-NODE · Power Plant | state dimension: 25 -> 12 | Helped |
| Use linear B-NODE β=0.01 instead of eDMDc | Linear surrogate · Power Plant | state dimension: 45 -> 25 | Helped |
| Use linear B-NODE β=0.01 instead of DMDc | Linear surrogate · Small Grid | RMSE: 0.60 -> 0.48 | Helped |
Surprising findings
- Vanilla Latent ODEs with inputs fail to achieve state reduction, because the latent distribution is only placed on the initial state.
- Adding a Latent NODE with beta=0.0 (no bottleneck control) on the power plant gave RMSE 18.2, far worse than the other models.
Limitations
- Dynamic-variance variant is sensitive to ODE stability and needs careful learning-rate tuning.
- Hyperparameter choices such as beta and the KL threshold of 0.1 for counting active dimensions are heuristic.
- Requires stable ODE training via growing horizon or multiple shooting strategies.
Experimental setup
Datasets: discretized heat flow (16 cells, simulated) thermal power plant simulation model (102 states, 6 outputs) MuJoCo Water Hammer Small Grid CFD model analytic nonlinear 2D system
Benchmarks: discretized heat flow power plant surrogate MuJoCo Water Hammer Small Grid
Metrics: RMSE maximum error state dimension computation time ODE function evaluations
Baselines: SS-NODE Latent NODE Truncated Balanced Realization eDMDc DMDc
Builds on: Neural ODE Latent ODE beta-VAE Truncated Balanced Realization DMDc eDMDc
Closest papers by meaning
- ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence (NeurIPS 2024 · similarity 0.87)
- Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs (ICLR 2024 · similarity 0.87)
- KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis (ICML 2025 · similarity 0.87)
- Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics (NeurIPS 2026 · similarity 0.86)
- Probabilistic Decomposed Linear Dynamical Systems for Robust Discovery of Latent Neural Dynamics (NeurIPS 2024 · similarity 0.86)
- Closing the Approximation Gap in Simulation-free Latent SDEs (NeurIPS 2026 · similarity 0.85)
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