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Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Data
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Proposes three stable Neural SDE classes (Langevin-type, linear noise, geometric) with controlled-path drift, proves existence/uniqueness and distribution-shift robustness bounds, and evaluates on PhysioNet, Speech Commands and 30 UEA/UCR datasets under missingness.
Problem
Naive Neural SDEs fail to train or are unstable on irregular, partially observed time series, and their robustness under distribution shift from missing data is not understood.
Approach
Design drift and diffusion functions from well-defined SDE families (Langevin-type with state-independent diffusion, linear multiplicative noise, geometric SDE with nonnegative absorbing states), add a controlled path (Neural CDE-style) into the drift, and prove existence/uniqueness and Wasserstein-stability bounds under input perturbations.
Key results
- On PhysioNet Mortality interpolation, Neural LSDE reaches MSE 3.799e-3 at 50% observed vs 4.139e-3 for mTAND-Full.
- On 30 UEA/UCR datasets at 70% missing, Neural LNSDE achieves 0.703 accuracy with average rank 4.2, the best among compared methods.
- On Speech Commands, Neural LSDE achieves 0.927 accuracy, above LEAP at 0.922, but uses more memory (1187 MB vs 391 MB).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a stable Neural LSDE or Neural LNSDE instead of mTAND-Full | PhysioNet Mortality interpolation · PhysioNet Mortality, 50%–90% observed | MSE (× 10^-3): 4.139 ± 0.029 -> 3.799 ± 0.055; 4.798 ± 0.036 -> 3.111 ± 0.076 | Helped |
| Use Neural LSDE or Neural LNSDE instead of ANCDE | PhysioNet Sepsis with OI · PhysioNet Sepsis with OI | AUROC: 0.900 ± 0.002 -> 0.911 ± 0.002 | Helped |
| Use Neural LSDE or Neural LNSDE instead of GRU-D | PhysioNet Sepsis without OI · PhysioNet Sepsis without OI | AUROC: 0.850 ± 0.013 -> 0.881 ± 0.002 | Helped |
| Use Neural LSDE instead of LEAP | Speech Commands classification · Speech Commands | accuracy: 0.922 ± 0.002 -> 0.927 ± 0.004 | Helped |
| Use Neural LNSDE instead of Neural CDE | UEA/UCR 30-dataset classification · 30 datasets, regular and 30%, 50%, and 70% missing | accuracy: 0.709 (0.061) -> 0.727 (0.047); 0.706 (0.073) -> 0.723 (0.050); 0.696 (0.064) -> 0.717 (0.054); 0.665 (0.072) -> 0.703 ( | Helped |
| Add a controlled path to the drift (ablation) | Neural LNSDE · UEA/UCR 30 datasets, all datasets | accuracy: 0.517 (0.064) -> 0.717 (0.051) | Helped |
| Use nonlinear neural-network diffusion instead of affine diffusion (ablation) | Neural LNSDE with controlled path · UEA/UCR 30 datasets, all datasets | accuracy: 0.675 (0.066) -> 0.717 (0.051) | Helped |
| Add a controlled path and nonlinear diffusion to the naive Neural SDE (ablation) | Neural SDE · UEA/UCR 30 datasets, all datasets | accuracy: 0.516 (0.064) -> 0.688 (0.063) | Helped |
| Replace affine diffusion with nonlinear neural-network diffusion (ablation) | Neural SDE with controlled path · UEA/UCR 30 datasets, all datasets | accuracy: 0.643 (0.068) -> 0.688 (0.063) | Helped |
| Add a controlled path to the drift (ablation) | Neural LSDE · UEA/UCR 30 datasets, all datasets | accuracy: 0.528 (0.060) -> 0.694 (0.056) | Helped |
| Use nonlinear neural-network diffusion instead of affine diffusion (ablation) | Neural LSDE with controlled path · UEA/UCR 30 datasets, all datasets | accuracy: 0.639 (0.064) -> 0.694 (0.056) | Helped |
| Add a controlled path to the drift (ablation) | Neural GSDE · UEA/UCR 30 datasets, all datasets | accuracy: 0.520 (0.060) -> 0.703 (0.063) | Helped |
Surprising findings
- The naive Neural SDE with non-Lipschitz or Hölder diffusion, or a neural-network diffusion, performs poorly and is unstable in training, so diffusion design matters a lot.
- Proposed Neural SDEs are less memory-efficient than CDE-based methods.
Limitations
- Proposed methods use more memory than CDE-based models
- Theoretical robustness bounds rely on assumptions such as Lipschitz and linear-growth conditions on neural networks
Experimental setup
Datasets: PhysioNet Mortality PhysioNet Sepsis Speech Commands MuJoCo UEA and UCR time series archives (30 datasets)
Benchmarks: PhysioNet Mortality PhysioNet Sepsis Speech Commands UEA/UCR 30 datasets with 30/50/70% missing
Metrics: mean squared error AUROC accuracy average rank cross-entropy loss memory usage
Baselines: RNN LSTM GRU GRU-D mTAN Neural CDE ANCDE LEAP Latent SDE Neural SDE
Builds on: Neural ODE Neural CDE Latent SDE
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