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From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems
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Proposes Fourier NODEs (FNODEs), a simulation-free method that estimates temporal and spatial gradients from noisy data via Fourier analysis, trains an MLP to match the gradient vector field, and augments data with the trained model in a feedback loop.
Problem
Standard neural ODEs for modeling complex dynamical systems are computationally expensive (simulation during training, stiff dynamics, adjoint sensitivity) and prone to local optima, especially with noisy and limited time-series data.
Approach
Replace ODE-solver-based training with direct flow matching: estimate temporal derivatives (and spatial derivatives for PDEs) with truncated Fourier series, regress the network's vector field onto these estimates, and iteratively augment the data with the trained model's predictions so that denser sampling improves the Fourier gradient estimates.
Key results
- On the 2-D parametric ODE, FNODE reached MSE 0.049 with 10 training samples and trained in 62.4 s versus 9632.2 s for NODE (Dopri5).
- Training time on PDE systems was 65-283 s for FNODEs versus 2910-21747 s for NODEs, i.e. over 10x faster as claimed.
- On NS with N_tr=1000 and no noise, FNODEs achieved MSE 0.083 versus 3.61 for NODEs and 2.07 for MP-PDE; PINN (0.31) was competitive but requires known equations and retraining per test sample.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Fourier gradient-flow matching instead of standard NODE training | 2-D parametric ODE · 2-D parametric ODE | training time: 9632.2 -> 62.4 s | Helped |
| Use Fourier gradient-flow matching instead of standard NODE training | FNODEs · KDV, DR, KS, and NS | training time | Helped |
| Use FNODEs instead of baselines | KDV, DR, KS, and NS · PDE prediction, N_tr=1000 and σ_sd=0 | prediction MSE: KDV: 0.45 -> 0.023; DR: 0.58 -> 0.016; KS: 1.69 -> 0.11; NS: 0.92 -> 0.083 | Helped |
| Use FNODEs instead of PINN | KDV, DR, KS, and NS · PDE prediction, N_tr=10 and σ_sd=0 | prediction MSE: KDV: 0.17 -> 0.23; DR: 0.19 -> 0.16; KS: 0.24 -> 0.28; NS: 0.31 -> 0.29 | Mixed |
| Use FNODEs instead of PINN | KDV, DR, KS, and NS · PDE prediction, N_tr=1000 and σ_sd=5% | prediction MSE: KDV: 0.17 -> 0.14; DR: 0.19 -> 0.12; KS: 0.24 -> 0.46; NS: 0.31 -> 0.11 | Mixed |
| Use data augmentation feedback (ablation) | FNODEs · Polar motion | prediction performance | Helped |
| Use data augmentation feedback (ablation) | FNODEs · Parametric ODE system | prediction error | Helped |
| Increase feedback data from 0 to 200 and 600 points (ablation) | FNODEs · Parametric ODE system | prediction error | Helped |
| Increase Fourier cutoff frequency K beyond the recoverable Nyquist range (ablation) | FNODEs · 2-D parametric ODE | prediction performance | No clear effect |
| Increase sampling points and usable cutoff frequency through augmentation (ablation) | FNODEs · Parametric ODE system | prediction error | Helped |
| Train with lower spatial resolution and predict at higher resolution (ablation) | FNODEs · DR PDE | prediction performance | Helped |
| Add Fourier-estimated spatial derivatives as neural-network inputs | FNODEs · PDE systems | prediction MSE | Helped |
Surprising findings
- PINN, which uses the known governing equations, performed comparably to or better than FNODEs on several PDE settings (e.g. KDV, DR, KS with 10 samples), and FNODEs were not best under noise on the ODE test (0.083 at 5% noise vs 0.134 for Mid is a gain, but 0.184 at 10% noise).
- The 'Mid' central-difference baseline achieved competitive results, suggesting the gain comes mainly from the gradient-matching formulation.
Limitations
- Fourier-based gradient estimation becomes difficult for more complex systems with higher cutoff frequency requirements.
- Inefficiency similar to NODEs on high-dimensional systems; prediction may be slower than neural operators such as DeepONet and FNO.
- Irregular sampling requires NDFT/NFFT and is not fully evaluated; the method assumes the control input is known.
Experimental setup
Datasets: synthetic parametric ODE data (GRF controls) KDV, DR, KS, NS simulated PDE data Lorenz63 with GRF-sampled rho(t) polar motion EOP time series (IERS) with EAM features
Benchmarks: 2-D parametric ODE KDV DR KS NS (Navier-Stokes) Lorenz63 polar motion prediction
Metrics: prediction MSE training time (s) MAE
Baselines: NODE (Euler) NODE (Dopri5) DeepONet FNO PDE-Net PINN DeepAR MP-PDE Mid (central difference)
Builds on: neural ODE DeepONet Fourier Neural Operator PDE-Net PINN
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