Structured Neural SDEs for Functional Calibration

Francesco Piatti; Andrea Iannucci; Thomas Cass

cs.LGSep 28, 2026

Structured Neural SDEs for Functional Calibration

Authors: Francesco Piatti, Andrea Iannucci, Thomas Cass

Organizations: Department of Mathematics, Imperial College London

Abstract

Neural Stochastic Differential Equations (Neural SDEs) provide flexible continuous-time generative models, but generic neural drift and diffusion networks are costly to simulate on long horizons and can give unstable gradients when the training signal is a path functional rather than a pointwise observation. We introduce SLiSDE, a family of Neural SDE models built from structured linear stochastic layers. Parallel-in-time simulation is obtained at the layer level, while expressivity is recovered by gated in-flow stacking: previous-layer paths modulate the next layer's latent flow through learned gates. For functional calibration tasks in which rare paths dominate the loss, we add an optional Girsanov tilt that acts as a learned importance sampler with an exact likelihood-ratio correction. We prove well-posedness, a discretisation error bound, validity of the change of measure, and a universality result: the terminal laws of the gated stack are dense in the space of square-integrable laws. Experiments on functional calibration benchmarks show that the structured model outperforms fully neural SDE baselines while retaining parallel-time simulation and stable importance weights.

Figures & tables

	Diagonal	Block-diagonal	Dense	NeuralSDE
Recurrent cost	$O(dT)$	$O(n_{b}b_{s}^{2}T)$	$O(d^{2}T)$	$O(d^{2}T)$
Scan cost	$O(d\log T)$	$O(n_{b}b_{s}^{3}\log T)$	$O(d^{3}\log T)$	—

Table 1: Computational cost of composing affine transition maps over

T

time steps. Here

d

is the latent dimension,

b_{s}

is the block size, and

n_{b}=d/b_{s}

is the number of blocks. Typically

n_{b}\ll d

Model	Params	Loss ( $10^{-4}$ )	Barrier MAE ( $10^{-2}$ )	ms / epoch
Neural SDE – $L=2$	124 K	$1.557\pm 0.206$	$1.058\pm 0.110$	91.7
SLiCE – $L=2$	142 K	$4.690\pm 0.423$	$2.168\pm 0.102$	171.8
SLiSDE gated in-flow – $L=2$	103 K	$\mathbf{0.852\pm 0.168}$	$\mathbf{0.658\pm 0.054}$	72.9
Neural SDE – $L=3$	157 K	$3.024\pm 0.341$	$1.637\pm 0.120$	110.4
SLiCE – $L=3$	213 K	$5.006\pm 0.914$	$2.250\pm 0.252$	248.2
SLiSDE gated in-flow – $L=3$	74 K	$\mathbf{0.659\pm 0.059}$	$\mathbf{0.672\pm 0.057}$	68.9

Table 2 : toy results without Girsanov tilt.

L

is the number of stacked layers; each row is the tuned cell of the search grid for that family and depth, retrained under the common protocol of Appendix D . We report parameter count, calibration loss, threshold-crossing error, and time per epoch.

Model	Params	Loss ( $10^{-4}$ )	Far put ( $10^{-6}$ )	ms / epoch
Neural SDE – $L=2$	79 K	$1.972\pm 0.117$	$9.87\pm 2.47$	91.4
SLiCE – $L=2$	142 K	$1.605\pm 0.149$	$\mathbf{8.14\pm 1.69}$	169.9
SLiSDE gated in-flow – $L=2$	90 K	$\mathbf{1.552\pm 0.070}$	$9.40\pm 2.30$	79.7
Neural SDE – $L=3$	157 K	$1.908\pm 0.180$	$8.20\pm 1.42$	114.4
SLiCE – $L=3$	213 K	$1.560\pm 0.086$	$6.41\pm 1.33$	246.9
SLiSDE gated in-flow – $L=3$	135 K	$\mathbf{1.401\pm 0.244}$	$\mathbf{4.07\pm 0.62}$	129.1

Table 3 : dax results without Girsanov tilt.

L

is the number of stacked layers; each row is the tuned cell of the search grid for that family and depth, retrained under the common protocol of Appendix D . We report parameter count, held-out loss, held-out far-put (OTM) loss, and time per epoch.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Params	Loss ( $10^{-4}$ )	Far put ( $10^{-6}$ )	ms / epoch
Neural SDE – $L=2$	79 K	$11.90\pm 0.11$	$135.4\pm 2.9$	94.1
SLiCE – $L=2$	142 K	$\mathbf{6.43\pm 0.12}$	$\mathbf{81.4\pm 4.6}$	170.1
SLiSDE gated in-flow – $L=2$	18 K	$6.44\pm 0.40$	$83.7\pm 4.8$	44.3
Neural SDE – $L=3$	157 K	$10.29\pm 0.13$	$136.0\pm 2.8$	109.7
SLiCE – $L=3$	213 K	$6.53\pm 0.26$	$82.9\pm 3.6$	248.4
SLiSDE gated in-flow – $L=3$	34 K	$\mathbf{6.29\pm 0.24}$	$\mathbf{70.1\pm 6.6}$	70.5

Appendix

Table 4 : spx option-surface results without Girsanov tilt, two and three layers. Same masks and protocol as Table 3 ; best cell of the search grid per family and depth, seven seeds, mean

\pm

standard error. The narrower three-layer SLiCE cell diverged on every seed and is excluded.

Noise type (block size $8$ )
Variant	Params	Loss	Far put	ms / epoch
multiplicative	$0.99\times$	$1.05\times$	$\mathbf{0.97\times}$	$0.93\times$
additive	$0.46\times$	$\mathbf{0.95\times}$	$1.69\times$	$\mathbf{0.78\times}$
general (both; reference)	$1.00\times$	$1.00\times$	$1.00\times$	$1.00\times$
Block size (general noise)
$b_{s}=1$ (diagonal)	$0.46\times$	$0.95\times$	$1.45\times$	$\mathbf{0.60\times}$

Appendix

Table 5 : Structural ablations on dax , each varying one factor of the two-layer gated in-flow configuration (general noise, block size

8

). Every entry is the ratio of the seven-seed mean of the variant to that of the reference configuration; the total-loss differences within each block are within one standard error. Top: noise type of the structured layers. Bottom: block size of the block-diagonal structure.

$B$	$T$	SLiSDE (seq.)	Neural SDE	SLiCE	speed-up
64	512	9.5	54.0	10.1	5.7 $\times$
64	2048	35.8	212.0	42.8	5.9 $\times$
64	8192	117.1	838.9	156.9	7.2 $\times$
256	2048	79.7	257.1	155.2	3.2 $\times$
1024	512	69.0	89.4	155.0	1.3 $\times$
1024	2048	258.9	359.5	OOM	1.4 $\times$

Appendix

Table 6: Wall-clock time per training step (ms; forward plus backward, post-compilation, median of five repeats) as a function of batch size

B

and horizon

T

for SLiSDE (

136

K parameters, sequential evaluation), the Neural SDE of the same parameter count (

137

K) and SLiCE (

142

K). Speed-up is Neural SDE over SLiSDE; OOM: exceeds the

42.6

GiB of the GPU.

$B$	$T$	SLiSDE fwd (seq.)	SLiSDE fwd (scan)	NSDE fwd	SLiSDE fwd+bwd	NSDE fwd+bwd
64	512	5.7	4.1	15.3	9.5	54.0
64	2048	14.9	15.8	50.6	35.8	212.0
64	8192	48.4	50.4	202.6	117.1	838.9
256	2048	32.5	44.6	61.0	79.7	257.1
1024	512	27.6	39.9	23.8	69.0	89.4
1024	2048	89.9	OOM	84.2	258.9	359.5

Appendix

Table 7 : Wall-clock time (ms, post-compilation, NVIDIA RTX 6000 Ada) of forward passes and full training steps as a function of batch size and horizon, including the chunked associative scan (best of chunk sizes

16

64

256

); the Neural SDE has the same parameter count as SLiSDE (

137

K against

136

K).

	KS (terminal)	$W_{1}$ (terminal)	KS (running max)	$W_{1}$ (running max)
Neural SDE ( $L=2$ )	$0.019\pm 0.002$	$\mathbf{0.020\pm 0.002}$	$0.059\pm 0.002$	$0.047\pm 0.002$
SLiCE ( $L=2$ )	$0.022\pm 0.002$	$0.031\pm 0.003$	$0.081\pm 0.003$	$0.067\pm 0.003$
SLiSDE ( $L=2$ )	$\mathbf{0.015\pm 0.001}$	$0.023\pm 0.002$	$\mathbf{0.036\pm 0.002}$	$\mathbf{0.037\pm 0.002}$

Appendix

Table 8 : Distances between the learned and the ground-truth laws of the terminal value and of the running maximum on toy (

2\times 10^{4}

model paths; best two-layer cell of each family; seven seeds, mean

\pm

s.e.).

Arm	paths / step	Loss ( $10^{-4}$ )	Far put ( $10^{-6}$ )	Far call ( $10^{-6}$ )	$\ESS/n_{q}$ put / call	$\KL$ put / call
Vanilla	$1024$	$1.15\pm 0.12$	$0.93\pm 0.46$	$3.15\pm 0.33$	–	–
Untilted, $\rho=0$	$1024$	$1.18\pm 0.19$	$0.87\pm 0.42$	$3.62\pm 0.78$	–	–
Untilted, $\rho=0$	$2048$	$0.95\pm 0.13$	$0.81\pm 0.30$	$2.59\pm 0.49$	–	–
Tilt, combined far-tail estimate	$1024+2{\times}512$	$0.98\pm 0.10$	$0.82\pm 0.27$	$\mathbf{2.03\pm 0.78}$	$0.05$ / $0.09$	$2.9$ / $1.5$
Tilt, tilted batch only	$1024+2{\times}512$	$\mathbf{0.88\pm 0.10}$	$\mathbf{0.48\pm 0.08}$	$3.25\pm 1.04$	$0.05$ / $0.12$	$3.0$ / $1.4$

Appendix

Table 9 : dax with the Girsanov tilt (seven seeds, mean

\pm

standard error). The tilt study is trained and evaluated on the strikes that lie within the quoted range at each expiry, so its absolute values are not comparable with Table 3 , whose protocol it otherwise follows. Far-tail losses are held-out MSEs on the far strikes; every arm is evaluated by plain Monte Carlo under its reference law with

32\,768

paths. The tilted arms use

\rho=0

, the cross-entropy controller with a

6

-nat KL wall and the initial-state shift;

\ESS/n_{q}

and

\KL(\mathbb{Q}\|\mathbb{P})

(nats) are end-of-training values of the put / call controllers; time per epoch

130

130

229

303

295

ms.

Estimator ( $n=1024$ ), rel. RMSE	$p\approx 10^{-2}$	$p\approx 10^{-3}$	$p\approx 10^{-4}$
Plain MC	$0.320\pm 0.005$	$1.051\pm 0.019$	$3.39\pm 0.39$
Plain MC, cost-matched ( $\times 1.18$ )	$0.278\pm 0.009$	$0.934\pm 0.047$	$3.15\pm 0.37$
Girsanov tilt, IS	$\mathbf{0.156\pm 0.018}$	$\mathbf{0.394\pm 0.104}$	$\mathbf{0.785\pm 0.184}$
Girsanov tilt, SNIS	$0.274\pm 0.021$	$0.789\pm 0.106$	$1.88\pm 0.42$
Share of plain-MC runs returning exactly $0$	0%	34%	90%

Appendix

Table 10: Rare-event estimation on toy : relative RMSE of each estimator at a budget of

n=1024

paths per estimate (cost-matched plain Monte Carlo receives

1.18\,n

paths), over

100

replications for each of seven independently calibrated models; mean

\pm

standard error over the models.

	rel. RMSE	$n=256$	$n=1024$	$n=4096$
$p\approx 10^{-2}$	Plain MC	$0.645\pm 0.017$	$0.320\pm 0.005$	$0.163\pm 0.005$
	Plain MC, cost-matched	$0.560\pm 0.012$	$0.278\pm 0.009$	$0.151\pm 0.003$
	Tilt, IS	$\mathbf{0.411\pm 0.084}$	$\mathbf{0.156\pm 0.018}$	$0.140\pm 0.033$
	Tilt, IS truncated	$0.391\pm 0.070$	$0.156\pm 0.018$	$\mathbf{0.119\pm 0.017}$
	Tilt, SNIS	$0.645\pm 0.131$	$0.274\pm 0.021$	$0.197\pm 0.034$
	plain MC returning $0$	$8\%$	$0\%$	$0\%$

Appendix

Table 11: Rare-event estimation on toy : relative RMSE for three budgets and three nominal probabilities, over

100

replications for each of seven independently calibrated models (mean

\pm

standard error over the models); cost-matched plain Monte Carlo receives

1.18\,n

paths; the last row of each block is the share of plain-Monte-Carlo replications with no path above the threshold.

Controller	Loss ( $10^{-4}$ )	Far put ( $10^{-6}$ )	Far call ( $10^{-6}$ )	$\ESS/n_{q}$ put / call
CE, $\kappa=6$ , combined (7 seeds)	$0.98\pm 0.10$	$0.82\pm 0.27$	$2.03\pm 0.78$	$0.05$ / $0.09$
CE, $\kappa=6$ , tilted batch only (7 seeds)	$0.88\pm 0.10$	$0.48\pm 0.08$	$3.25\pm 1.04$	$0.05$ / $0.12$
CE, $\kappa=2$ , combined (3 seeds)	$1.13\pm 0.20$	$0.59\pm 0.01$	$2.46\pm 1.03$	$0.06$ / $0.08$
SNIS-variance, combined (3 seeds)	$1.18\pm 0.35$	$1.18\pm 0.33$	$2.71\pm 1.10$	$1.00$ / $0.99$

Appendix

Table 12: Controller ablations on dax far-tail calibration (

\rho=0

, protocol of Table 9 ; mean

\pm

standard error over seeds).

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EPFL, Imperial College · EPFL

Stochastic Differential Equations Stochastic Dynamics Kolmogorov-Arnold Networks

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Neural stochastic differential equations (SDEs) have emerged as powerful tools for learning noisy or stochastic dynamics directly from data; however, existing approaches largely assume uncoupled and continuous noise, limiting their applicability to realistic stochastic drivers, and often scale poorly in time, requiring expensive autoregressive training. To address these limitations, we propose Neural Kolmogorov Equations (NKEs), a deterministic, infinite-dimensional reformulation of Neural SDEs based on the Kolmogorov Forward equation, transforming the learning problem from modelling individual stochastic trajectories to modelling the evolution of probability densities. NKEs learn general Lévy-type stochastic forcing directly through the operator structure of the KFE, and enable parallel-in-time training via a Lagrangian Galerkin projection and operator splitting. We evaluate NKEs on several stochastic benchmarks, including systems with coupled noise and jump processes, and verify that NKEs provide flexible models that accurately recover deterministic and stochastic dynamics with competitive predictive accuracy and improved training efficiency. Code and pretrained models will be released.

Abstract

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