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Statistical Spatially Inhomogeneous Diffusion Inference
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Proves fast-rate (near-minimax) statistical guarantees for sparse-ReLU neural estimators of drift and spatially-inhomogeneous diffusion tensor from a single ergodic, beta-mixing trajectory, and validates the scaling on a 2D periodic SDE.
Problem
Estimating drift and spatially-inhomogeneous diffusion tensor of an SDE from discretely observed, temporally correlated single-trajectory data, with statistical convergence guarantees beyond the i.i.d. setting.
Approach
Neural (sparse ReLU) least-squares estimators of drift and diffusion tensor are analyzed under exponential ergodicity and beta-mixing; sub-sampling the trajectory into approximately i.i.d. sub-sequences combined with local Rademacher complexity yields fast rates N^{-2s/(2s+d)} log^3 N plus discretization and ergodic terms.
Key results
- Theory: drift error bound T^{-2s/(2s+d)} log^3 T + tau and diffusion error bound N^{-2s/(2s+d)} log^3 N + tau + log^2 N / T, matching the minimax nonparametric rate up to log factors.
- Numerical experiments on a 2D SDE show test error decaying roughly exponentially in N for fixed tau=1e-3 and fixed T=1e3, consistent with the predicted N^{-1}+tau+T^{-1} scaling.
- Error decay slows once N is large because the discretization bias tau dominates.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase the number of observations N while keeping the time step τ=10^{-3} fixed | Neural diffusion estimator · Two-dimensional synthetic diffusion, fixed τ=10^{-3} | test error | Helped |
| Increase the number of observations N while keeping the time horizon T=10^3 fixed | Neural diffusion estimator · Two-dimensional synthetic diffusion, fixed T=10^3 | test error | Helped |
Limitations
- Restricted to time-homogeneous coefficients on a periodic torus domain; time-dependent SDEs left for future work.
- Only a single 2D numerical example is reported; empirical validation is limited.
- Rate includes extra log factors and a tau discretization bias term.
Experimental setup
Datasets: synthetic 2D SDE trajectory simulated via Euler-Maruyama
Benchmarks: synthetic 2D periodic SDE with f(x)=1+0.5cos(2pi(x1+x2))
Metrics: test loss (estimation error) vs N, tau, T scaling
Baselines: Euler-Maruyama data generation (no estimator baseline reported)
Builds on: local Rademacher complexity beta-mixing generalization bounds sparse ReLU neural network estimation theory
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