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Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions

ICLR 2025 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural ODE training and analysis · 0 citations (OpenAlex)

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Proposes Finite Dimensional Matching (FDM), a strictly proper scoring rule for continuous Markov processes built from two-time joint distributions, used to train Neural SDEs with O(D) per-epoch cost instead of signature-kernel PDE solves.

Problem

Training Neural SDEs as generative models of stochastic processes: existing methods rely on GAN adversarial training (unstable) or signature kernel / truncated signature scoring rules that require solving Goursat PDEs (quadratic cost in discretization steps) or are not strictly proper.

Approach

Shows that any strictly proper scoring rule on R^{2d} (two-time joint distributions) can be averaged over uniformly sampled time pairs to give a strictly proper scoring rule for continuous Markov processes; with an RBF kernel this yields a cheap MMD-style objective trained by standard ERM.

Key results

• On metal prices (64 timestamps), FDM achieves average KS scores around 0.112-0.118 versus about 0.122-0.144 for SigKer and 0.27-0.83 for TruncSig and SDE-GAN.
• On 32-dimensional rough Bergomi, TruncSig runs out of GPU memory while FDM attains KS of 0.117 at t=6 compared with 0.120 for SigKer.
• SigKer hits a thread limit at 1024 timestamps on exchange rates, while FDM remains trainable and outperforms TruncSig and SDE-GAN there.

Measured effects reported in the paper

Surprising findings

• On the 16-dimensional rough Bergomi model, SigKer slightly outperforms FDM at early timestamps (e.g., KS .112 vs .113 at t=6), and SigKer beats FDM on some earliest time points of the 256-timestep EUR/USD table.

Limitations

• Theory requires continuity and Markov property; non-Markov processes (e.g., hidden Markov) are not covered.
• Experiments use only the RBF kernel with unit bandwidth and fixed step counts; sensitivity to kernel choice not explored in main text.
• Baselines are trained for different numbers of steps to normalize training time, which complicates direct comparison.

Experimental setup

Datasets: metal prices U.S. stock indices exchange rates (EUR/USD, USD/JPY) energy prices bonds rough Bergomi synthetic model

Benchmarks: metal prices U.S. stock indices exchange rates energy prices bonds rough Bergomi

Metrics: KS test statistic rejection rate at 5% significance wall-clock training time

Baselines: SigKer TruncSig SDE-GAN

Builds on: Neural SDE signature kernel scoring rule SDE-GAN latent SDE truncated signature

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