Learning Stochastic Thermodynamics Directly from Correlation and Trajectory-Fluctuation Currents

TL;DR

This work addresses data-driven inference of stochastic thermodynamics for Langevin-type systems by deriving loss functions directly from underlying dynamics using currents, rather than relying on TUR saturation. It shows that mean-squared-error losses can be recast as correlation-current estimates, enabling inference of average and trajectory-resolved entropy production, forces, and diffusion fields in overdamped, underdamped, and Markovian settings. The authors introduce higher-order loss functions to cope with mismatched time scales and partial observations, and demonstrate per-trajectory entropy production recovery in a bead-spring model driven far from steady state. This framework unifies dynamic inference with entropy-production estimation, offering a flexible, data-driven path to probe nonequilibrium thermodynamics beyond TUR limitations and with potential applications to real-time monitoring and quantum systems.

Abstract

Markedly increased computational power and data acquisition have led to growing interest in data-driven inverse dynamics problems. These seek to answer a fundamental question: What can we learn from time series measurements of a complex dynamical system? For small systems interacting with external environments, the effective dynamics are inherently stochastic, making it crucial to properly manage noise in data. Here, we explore this for systems obeying Langevin dynamics and, using currents, we construct a learning framework for stochastic modeling. Currents have recently gained increased attention for their role in bounding entropy production (EP) from thermodynamic uncertainty relations (TURs). We introduce a fundamental relationship between the cumulant currents there and standard machine-learning loss functions. Using this, we derive loss functions for several key thermodynamic functions directly from the system dynamics without the (common) intermediate step of deriving a TUR. These loss functions reproduce results derived both from TURs and other methods. More significantly, they open a path to discover new loss functions for previously inaccessible quantities. Notably, this includes access to per-trajectory entropy production, even if the observed system is driven far from its steady-state. We also consider higher order estimation. Our method is straightforward and unifies dynamic inference with recent approaches to entropy production estimation. Taken altogether, this reveals a deep connection between diffusion models in machine learning and entropy production estimation in stochastic thermodynamics.

Figures (4)

• Figure 1: Learning framework: Input data are discrete-time positions: $\{x(t), x(t+\Delta t), x(t+2\Delta t),\dots\}$ and, for underdamped systems, the estimated velocity $\widehat{v}(t) = [x(t+\Delta t) - x(t)]/\Delta t$ is also required. To infer local thermodynamic function $F$ at time $t$---such as force, local entropy production, and diffusion fields---we construct first- and second-order loss functions for training neural networks. First-order estimation relies on two consecutive data points, while second-order estimation uses three. Representative first-order loss functions are summarized in Table \ref{['tab:tablesummary']}.
• Figure 2: Entropy production inference with five beads-spring model. (a) The estimated relative entropy production $\hat{\Sigma}/\Sigma$ as a function of the number of paths using with different coarse steps. The left/right are the relative entropy results with the first/second order loss functions. (b) The $1-R^2$ score between trajectory's theoretical and estimated stochastic entropy productions as functions of number of paths. The left/right are the results with the first/second order loss functions. (c) The relative entropy production results at different temperature ratios with different types of $10^6$ coarse-grained trajectories. (d) The corresponding $1-R^2$ score between theoretical stochastic entropy production and estimated entropy production of validating trajectories. (e) Entropy production of six chosen trajectories from the bead-spring model at the temperature ratio $T_{h}/T_{c}=2$. The theoretical stochastic entropy productions are computed from the definition: the sum of change in the surprisal and the heat along the trajectory. The dashed line is the average entropy production as a function of time $t$. (f) The role of $\partial_{t}\log f$ in stochastic entropy production. The stochastic entropy production (red) and Stratonovich integral of $u$ (blue) along trajectories are shown. The dashed lines are average entropy production and $\int D^{-1}\cdot \boldsymbol{u} \circ d\boldsymbol{x}$.
• Figure 3: Mean entropy production and $R^2$ score between theoretical and estimated stochastic entropy production with initial distribution $\Sigma_{2}$ at $T_{h}/T_{c}=2$: (a) Relative entropy production as a function of the number of paths using different coarse steps. The left (right) plot is the relative entropy with the first- (second-)order loss functions. (b) $1-R^2$ score between trajectory’s theoretical and estimated stochastic entropy productions as functions of the number of paths. The left (right) plot is the result with the first- (second-)order loss functions.
• Figure 4: Mean entropy production and $R^2$ score between the theoretical and estimated stochastic entropy production with $\Sigma_{\text{NESS}}$ at $T_{h}/T_{c}=2$: (a) Relative entropy production as a function of the number of paths using different coarse steps. The left (right) plot gives the relative entropy results with the first- (second-)order loss functions. (b) $1-R^2$ score between trajectory’s theoretical and estimated stochastic entropy productions as functions of the number of paths. The left (right) plot gives the results with the first- (second-)order loss functions.