Deterministic particle flows for constraining stochastic nonlinear systems

Abstract

Devising optimal interventions for constraining stochastic systems is a challenging endeavour that has to confront the interplay between randomness and nonlinearity. Existing methods for identifying the necessary dynamical adjustments resort either to space discretising solutions of ensuing partial differential equations, or to iterative stochastic path sampling schemes. Yet, both approaches become computationally demanding for increasing system dimension. Here, we propose a generally applicable and practically feasible non-iterative methodology for obtaining optimal dynamical interventions for diffusive nonlinear systems. We estimate the necessary controls from an interacting particle approximation to the logarithmic gradient of two forward probability flows, evolved following deterministic particle dynamics. Applied to several biologically inspired models, we show that our method provides the necessary optimal controls in settings with terminal-, transient-, or generalised collective-state constraints and arbitrary system dynamics.

Neural Lagrangian Schrödinger Bridge

“…Moreover, designing the Lagrangian enables us to flexibly incorporate prior knowledge about the target system into the model along with the principle of least action. Our Lagrangian-based regularization also treated the biological constraints proposed by Tong et al [25] and Maoutsa and Opper [18] in a unified manner. In experiments, Figures 2b and 3c, and Table 3 indicate that the prior knowledge introduced by the Lagrangian is useful to estimate the trajectories of individual samples with stochastic behavior.…”

Transport with Support: Data-Conditional Diffusion Bridges

“…Moreover, neural stochastic control has been studied in Zhang et al (2022). An optimal control problem with both initial and terminal distribution constraints and a fixed path constraint has been studied in Maoutsa et al (2020) and Maoutsa & Opper (2021), where particle filtering is applied to continuous path constraints but the boundary constraints are defined by a single point. Furthermore, the combination of Schrödinger bridges and state-space models has been studied by Reich (2019), in a setting where Schrödinger bridges are applied to the transport problem between filtering distributions.…”

“…The learned backward drift b l,φ can be interpreted as an analogy of the backward drift in Maoutsa & Opper (2021), connecting our approach to solving optimal control problems through Hamilton-Jacobi equations, see App. A.2 for an analysis of the backwards SDE and the control objective.…”

“…4 for modelling bird migration and wintering patterns. Recently, constrained optimal control problems have been explored by adding additional fixed path constraints (Maoutsa et al, 2020;Maoutsa & Opper, 2021) or modifying the prior processes (Fernandes et al, 2021). However, defining meaningful fixed path constraints or prior processes for the optimal control problems can be challenging, while sparse observational data are accessible in many real-world applications.…”

“…We perform the conditioning by leveraging the iterative pass idea from the Iterative Proportional Fitting procedure (IPFP) (Kullback, 1968;De Bortoli et al, 2021) and applying differentiable particle filtering (Reich, 2013;Corenflos et al, 2021) within the outer loop. Integrating sequential Monte Carlo methods (e.g., Doucet et al, 2001;Chopin & Papaspiliopoulos, 2020) into the IPFP framework in such a way is non-trivial and can be understood as a novel iterative version of the algorithm by Maoutsa & Opper (2021) but with more general marginal constraints and additional path constraints defined by data.…”