LECBC-LECASBC-LEC
Each method trains on sampled noise paths and PDE residuals only. The shaded surface is the trained network's expected solution E[u]; toggle the noise paths (single-path solutions of the same equation) and the true mean on top of it, or colour the surface by its error. Drag to rotate.
Each problem is a linear heat-type equation on the unit square, driven by one of the same nine noise drivers. Each is solved for its mean E[u]. They differ in where the noise ξ(t) enters, and that decides how hard the mean is to learn. All three start from the first Dirichlet mode G(x) = Π sin(πxi), with zero walls, t ∈ [0, 1] and ν ∈ [0.01, 0.1].
Only the additive problem has a closed-form mean, because the forcing and the initial condition both lie along G. Only the one-time moments of ξ enter it, which makes LEC a clean benchmark. The other two are scored against Monte Carlo means of per-path solves, which are never used in training.
The surface above is d = 2, at one parameter draw per driver: the 70th percentile of the driver's sampled spread. For LEC and BC-LEC the axes are (x₁, x₂). For ASBC-LEC they are (x, z), and the space–time view slices at x₂ = ½, or at z = 0 for ASBC-LEC. The networks shown are the 10-paths-per-point cells. LEC is the polished network, BC-LEC is at its final adaptive order, and ASBC-LEC uses the centre and order its own error estimate picked. Its correction scales were refitted after training from the training objectives. They reproduce each run's reported error for that draw to within a few hundredths of a percent.
Each driver is sampled from its exact marginal. Its parameters are appended to the network input, so one model covers the whole parameter box. Faint lines are sample paths; the solid line is E[ξ(t)] with a ±1 sd band.
ξ = μt + σWt · μ 0.3 · σ 0.8
dξ = θ(μ − ξ)dt + σdW · θ 1 · μ 0.3 · σ 0.8
ξ = Σ Ji, Nt ~ Poi(λt) · λ 3 · J ~ N(0.3, 0.6²)
Each process is shaped by W(ξ) ∈ {ξ, e⁻ᵗξ, ξ²}, giving nine drivers. The same nine drive all three methods. Parameter boxes: μ ∈ [0.1, 0.5], σ ∈ [0.1, 1], θ ∈ [0.5, 2], λ ∈ [0.1, 5], μJ ∈ [0.1, 0.5], σJ ∈ [0.1, 1].
At every collocation point the network sees a fresh draw of ξ. The stochastic forcing ϑG(x)F(t, ξ) is pure speckle. Averaging the PDE residual over m draws before squaring cancels the noise in expectation, so the loss is minimized by E[u] and not by any single realization.
What a single-sample residual is fitted against.
ϑG(x)·E[F(t, ξ(t))], the target the loss collapses to.
RMS gap between the m-draw average and the expected forcing, over the unit square. Line: theory. Dot: the field shown above.
LEC's additive problem with fixed parameters, nine paths read off at the peak of G. Orange where a path is above E[u], aqua where it is below. The vertical line is the time shown in the heatmaps.
This chamber shows LEC's source, which needs no expansion. BC-LEC and ASBC-LEC run the same averaged-residual collapse once per order. Their sources are built from sampled path values and noise-free kernels, η(t)·ℬu₀ at order 1, so the identity still holds term by term. Chamber 04 shows how.
The collapse works because the stochastic source does not depend on the network. Put the noise in a coefficient and that breaks. The plain LEC loss then splits into a mean-coefficient residual plus a penalty on ℬm, so it converges to a regularised problem, which is neither the noise-free solution nor E[u].
The fix is to expand the solution around a reference coefficient, so that every term again solves a linear equation with a source that does not involve the unknown. BC-LEC and ASBC-LEC both build that expansion as a Dyson series. They differ in how far it can be pushed.
Each mn has a source that needs un−1 along the path. The kernels write that term as an integral of η(s₁)⋯η(sn) against a noise-free field, so no per-path solve is ever needed. The driver's moments never appear in closed form. Products of path values from the sampler are unbiased samples of them.
Relative error of learned kernels against exact ones (scoring only), log scale. Each order applies νΔx to a learned network twice, and its Laplacian is about ten times less accurate than its values. The error grows 15–100× per order. A weak form slows the growth but does not stop it.
ASBC-LEC expands around a deterministic multiplier learned from samples of log a. One centre is geometric, eE W, and finite under any tail. The other is moment-matched, eE W + ½ Var W, and better for light tails. Both stacks are trained. For each parameter draw, the centre and order (0, 1 or 2) with the smallest training-only error estimate are kept.
| class | rule, first match wins |
|---|---|
| certified | estimate < 1%, the answer is accepted |
| flagged | series no longer asymptotic, or estimate not finite |
| learning-limited | truncation is small; more orders cannot help |
| needs order | truncation dominates and the series still shrinks |
Median relative L2 error (%) over 128 parameter draws against a reference mean, at t ∈ {0.25, 0.5, 0.75, 1}. Noise parameters and ν are network inputs, so each cell is one amortised network. Training sees sampled noise paths and PDE residuals only: no solution values, no pathwise solves, no noise moments.
| regime | outcome |
|---|---|
| Additive noise, LEC with polish | 0.18–0.81% at every paths budget, except compound Poisson ξ² at 2.4–3.6% |
| Multiplicative, lower-order term | under 1% wherever E[u] exists, given enough paths. Compound Poisson ξ reaches 0.35% at 256 paths. |
| Principal part, light tails | under 1% at the median, 4–20× better than the mean-coefficient solution |
| Principal part, Brownian ξ | about 1%, limited by truncation at N = 2 |
| Principal part, heavy tails | compound Poisson ξ: 4–5%. ξ²: fails at 31–38%, with 87% of draws flagged and none certified. |
| E[u] = ∞ | no target to learn; flagged |
In every failing regime the training-only estimate declines to certify, so the failure is visible without the solution. Dimensions 4, 6 and 8 are running on the cluster.
144 runs from the preprint: soft initial and boundary penalties and the original training schedule. Medians over forcing type and m, 12 runs per cell. These numbers are relative, not percent, and are not comparable with the table above.
@article{petursson2025chaos,
title = {Chaos into Order: Neural Framework for Expected Value Estimation of Stochastic Partial Differential Equations},
author = {P{\'e}tursson, {\'I}sak and {\'O}skarsd{\'o}ttir, Mar{\'i}a},
journal = {arXiv preprint arXiv:2502.03670},
year = {2025}
}