Interpretable Neural Approximation of Stochastic Reaction Dynamics with Guaranteed Reliability

TL;DR

DeepSKA is introduced, a neural framework that jointly achieves interpretability, guaranteed reliability, and substantial computational gains and offers a principled foundation for developing analogous methods for other Markovian systems, including stochastic differential equations.

Abstract

Stochastic Reaction Networks (SRNs) are a fundamental modeling framework for systems ranging from chemical kinetics and epidemiology to ecological and synthetic biological processes. A central computational challenge is the estimation of expected outputs across initial conditions and times, a task that is rarely solvable analytically and becomes computationally prohibitive with current methods such as Finite State Projection or the Stochastic Simulation Algorithm. Existing deep learning approaches offer empirical scalability, but provide neither interpretability nor reliability guarantees, limiting their use in scientific analysis and in applications where model outputs inform real-world decisions. Here we introduce DeepSKA, a neural framework that jointly achieves interpretability, guaranteed reliability, and substantial computational gains. DeepSKA yields mathematically transparent representations that generalise across states, times, and output functions, and it integrates this structure with a small number of stochastic simulations to produce unbiased, provably convergent, and dramatically lower-variance estimates than classical Monte Carlo. We demonstrate these capabilities across nine SRNs, including nonlinear and non-mass-action models with up to ten species, where DeepSKA delivers accurate predictions and orders-of-magnitude efficiency improvements. This interpretable and reliable neural framework offers a principled foundation for developing analogous methods for other Markovian systems, including stochastic differential equations.

Figures (38)

• Figure 1: The DeepSKA framework.a, Stochastic Reaction Networks (SRNs) are mechanistic models describing the time evolution of systems across the life sciences. In this context, functions $f$ of the random state are studied to capture some properties of the system of interest. A central goal is the estimation of expected outputs. b, The DeepSKA framework is composed of two complementary components: the Spectral Decomposition-based network (SDnet), which is an interpretable architecture to represent expected outputs; and two Deep Learning/Monte Carlo (DLMC) estimators, SSA with DeepCV and SSA with DeepIS, to provide provably reliable estimates of expected outputs.
• Figure 1: The Spectral Decomposition-based network (SDnet). The architecture is motivated by the spectral decomposition of $U$ in equation \ref{['eq:complex_decomposition_raw']}, which is approximated in equation \ref{['eq:complex_decomposition_expanded_approx']}. Specifically, $\hat{\sigma}_{\ell} \coloneqq \hat{a}_{\ell} + i \hat{b}_{\ell}$ serves as an approximation of a decay mode, $\hat{\gamma}_{\ell}(f) \coloneqq \hat{h}_{\ell}(f) + i \hat{g}_{\ell}(f)$ approximates a function coordinate, and $\hat{\phi}_{\ell} \coloneqq \hat{c}_{\ell} + i \hat{d}_{\ell}$ approximates an eigenfunction. Green: inputs. Purple: output. Orange: trainable or fixed variables. Blue: feedforward neural network. White: intermediate quantities.
• Figure 2: Components of the DeepSKA framework.a, The spectral decomposition of $U(x,f,t)$ leads to a representation which is used to define the Spectral Decomposition-based network (SDnet). The neural network is interpretable and its computations are transparent. In the schematic, inputs to SDnet are shown in green, outputs in purple, trainable variables in orange, and the feedforward neural network in blue. The architecture is trained with an extension of the Reinforcement Learning (RL) procedure of ref. gupta2021deepcme, whose loss is motivated by an almost sure relationship and is computed using simulated trajectories. b, The SSA with Deep Importance Sampling (SSA with DeepIS) and the SSA with Deep Control Variates (SSA with DeepCV) estimators combine Monte Carlo simulations with the trained SDnet. These Deep Learning/Monte Carlo (DLMC) estimators are reliable, with an estimation error which is expected to be lower than that of the standard SSA. c, We study nine Stochastic Reaction Networks (SRNs) in the DeepSKA framework. They are formally defined in section \ref{['supp_section:srns']} of the supplementary material. The presence of nonlinear and non-mass-action (MA) kinetics, as well as the species number, are used as a measure of the complexity of the networks. $^{*}$: results presented in sections \ref{['supp_section:nonlinear']} and \ref{['supp_section:linear']} of the supplementary material.
• Figure 2: The Poisson Spectral Decomposition-based network (P-SDnet). The architecture is motivated by the spectral decomposition of $F$ in equation \ref{['eq:poisson_spectral']}, which is approximated in equation \ref{['eq:poisson_spectral_approx']}. Green: inputs. Purple: output. Orange: fixed or trainable variables. Blue: feedforward neural network. White: intermediate quantities.
• Figure 3: Results for the toggle switch.a, Reaction graph of the system. The toggle switch consists of two mutually repressing proteins, $\mathbf{P_1}$ and $\mathbf{P_2}$. Each protein inhibits the production of the other through a Hill-type repression, and both species undergo degradation. b, Mean dynamics for different initial states and output functions obtained with the Spectral Decomposition-based network (SDnet) and the Stochastic Simulation Algorithm (SSA). Each plot shows the temporal evolution of the mean from time 0 to 60 for a given initial state and output function. Purple lines indicate SDnet predictions, and green bands show the mean and 95% confidence interval (CI) computed from 50 000 SSA samples. As in all other plots, solid lines correspond to the training interval $[0,20]$, and dashed lines indicate times beyond the training interval. Initial states vary horizontally from left to right, output functions vary vertically from top to bottom. The first output function corresponds to the first moment of protein $\mathbf{P_1}$ ($f_1(x) = x_1$), and the second to the second moment ($f_2(x) = (x_1)^2$). c, Mean dynamics under different initial states and output functions obtained with SSA with Deep Importance Sampling (SSA with DeepIS), SSA with Deep Control Variates (SSA with DeepCV), and SSA. The layout follows panel b. Curves for the Deep Learning/Monte Carlo (DLMC) estimators are computed from 200 samples and displayed with 95% CIs. d, Temporal evolution of the variance of the estimators for different initial states and output functions. The panel layout mirrors panels b and c. e, Mean-squared error (MSE) of the estimators at time $t=5$ for the initial state $[5,1]$ as a function of sample size. Reference values are computed from SSA with DeepCV using 100 000 samples. The squared error is averaged over 20 independent runs. f, Five sample paths obtained with SSA with DeepIS, SSA with DeepCV, and SSA. The red dot indicates the reference mean abundance at $t=60$, computed from 50 000 SSA samples.

Theorems & Definitions (17)

• Example B.1: Constitutive gene expression network thattai2001intrinsic
• Example B.2: Self-regulatory gene expression network gupta2022frequency
• Example B.3: Genetic toggle switch network gardner2000construction
• Example B.4: Susceptible–infected–recovered network thanh2015simulation
• Example B.5: Reference-based antithetic integral control of gene expression network briat2016antithetic
• Example B.6: Sensor-based antithetic integral control of gene expression network filo2023hidden
• Example B.7: Repressilator network elowitz2000synthetic
• Example B.8: Nonlinear conversion cascade network tang2023neural
• Example B.9: Linear conversion cascade with feedback network tang2023neural
• proof : Proof of equation \ref{['eq:poisson_spectral']}