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Time-Reversible Bridges of Data with Machine Learning

Ludwig Winkler

TL;DR

This work advances time-reversible modeling by learning dynamics under boundary constraints across three problem classes: deterministic boundary value problems in molecular dynamics, stochastic half-bridges for discrete jump processes (Ehrenfest), and full Schrödinger bridges between probability distributions. It introduces neural-network-based time-reversible integrators for MD trajectories, establishes a theoretical link between discrete Ehrenfest dynamics and continuous diffusion via the Ornstein-Uhlenbeck limit, and presents IPF-based learning of both forward and backward drifts for Schrödinger bridges with score-estimation surrogates. Key contributions include bi-directional neural networks for MD interpolation achieving near-quantum-accurate spatiotemporal reconstructions, a diffusion-model-inspired framework for discrete reverse-time learning, and a variational, score-based approach to learning full stochastic bridges without ground-truth drifts. The methods enable data-driven, high-fidelity interpolation and sampling across physics, chemistry, and biology, offering scalable tools for trajectory reconstruction, sample generation, and probabilistic transport between distributions. Overall, the thesis demonstrates that neural networks can faithfully learn time-reversible dynamics under complex boundary conditions, providing a versatile toolkit for solving otherwise intractable dynamical systems.

Abstract

The analysis of dynamical systems is a fundamental tool in the natural sciences and engineering. It is used to understand the evolution of systems as large as entire galaxies and as small as individual molecules. With predefined conditions on the evolution of dy-namical systems, the underlying differential equations have to fulfill specific constraints in time and space. This class of problems is known as boundary value problems. This thesis presents novel approaches to learn time-reversible deterministic and stochastic dynamics constrained by initial and final conditions. The dynamics are inferred by machine learning algorithms from observed data, which is in contrast to the traditional approach of solving differential equations by numerical integration. The work in this thesis examines a set of problems of increasing difficulty each of which is concerned with learning a different aspect of the dynamics. Initially, we consider learning deterministic dynamics from ground truth solutions which are constrained by deterministic boundary conditions. Secondly, we study a boundary value problem in discrete state spaces, where the forward dynamics follow a stochastic jump process and the boundary conditions are discrete probability distributions. In particular, the stochastic dynamics of a specific jump process, the Ehrenfest process, is considered and the reverse time dynamics are inferred with machine learning. Finally, we investigate the problem of inferring the dynamics of a continuous-time stochastic process between two probability distributions without any reference information. Here, we propose a novel criterion to learn time-reversible dynamics of two stochastic processes to solve the Schrödinger Bridge Problem.

Time-Reversible Bridges of Data with Machine Learning

TL;DR

This work advances time-reversible modeling by learning dynamics under boundary constraints across three problem classes: deterministic boundary value problems in molecular dynamics, stochastic half-bridges for discrete jump processes (Ehrenfest), and full Schrödinger bridges between probability distributions. It introduces neural-network-based time-reversible integrators for MD trajectories, establishes a theoretical link between discrete Ehrenfest dynamics and continuous diffusion via the Ornstein-Uhlenbeck limit, and presents IPF-based learning of both forward and backward drifts for Schrödinger bridges with score-estimation surrogates. Key contributions include bi-directional neural networks for MD interpolation achieving near-quantum-accurate spatiotemporal reconstructions, a diffusion-model-inspired framework for discrete reverse-time learning, and a variational, score-based approach to learning full stochastic bridges without ground-truth drifts. The methods enable data-driven, high-fidelity interpolation and sampling across physics, chemistry, and biology, offering scalable tools for trajectory reconstruction, sample generation, and probabilistic transport between distributions. Overall, the thesis demonstrates that neural networks can faithfully learn time-reversible dynamics under complex boundary conditions, providing a versatile toolkit for solving otherwise intractable dynamical systems.

Abstract

The analysis of dynamical systems is a fundamental tool in the natural sciences and engineering. It is used to understand the evolution of systems as large as entire galaxies and as small as individual molecules. With predefined conditions on the evolution of dy-namical systems, the underlying differential equations have to fulfill specific constraints in time and space. This class of problems is known as boundary value problems. This thesis presents novel approaches to learn time-reversible deterministic and stochastic dynamics constrained by initial and final conditions. The dynamics are inferred by machine learning algorithms from observed data, which is in contrast to the traditional approach of solving differential equations by numerical integration. The work in this thesis examines a set of problems of increasing difficulty each of which is concerned with learning a different aspect of the dynamics. Initially, we consider learning deterministic dynamics from ground truth solutions which are constrained by deterministic boundary conditions. Secondly, we study a boundary value problem in discrete state spaces, where the forward dynamics follow a stochastic jump process and the boundary conditions are discrete probability distributions. In particular, the stochastic dynamics of a specific jump process, the Ehrenfest process, is considered and the reverse time dynamics are inferred with machine learning. Finally, we investigate the problem of inferring the dynamics of a continuous-time stochastic process between two probability distributions without any reference information. Here, we propose a novel criterion to learn time-reversible dynamics of two stochastic processes to solve the Schrödinger Bridge Problem.

Paper Structure

This paper contains 84 sections, 4 theorems, 139 equations, 33 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Objective and Scope
  3. Outline
  4. List of Publications
  5. Theoretical Context and Fundamentals
  6. Neural Networks
  7. Ordinary Differential Equations
  8. Reverse Time Ordinary Differential Equations
  9. Stochastic Processes
  10. Stochastic Differential Equations
  11. The Fokker-Planck Equation
  12. Jump Processes
  13. Master Equation for Jump Processes
  14. Reverse Time Stochastic Processes
  15. Reverse Time Stochastic Differential Equations
  16. ...and 69 more sections

Key Result

Proposition 3.2.1

For a time-reversible dynamic $f(x_\tau, \tau)$ with the boundary conditions $x_t$ and $x_{t + \Delta t}$ on the time interval $[t, t + \Delta t]$, the solution $x_\tau$ at a time $\tau \in [t , t + \Delta t]$ is given by, with the time-dependent weighting

Figures (33)

  • Figure 1: In \ref{['cha:deterministicbvps']} we consider boundary value problems in a deterministic setting in which we learn the time-reversible, deterministic dynamics with neural networks from observed solutions. The boundary conditions $x_0$ and $x_T$ are deterministic.
  • Figure 2: In \ref{['cha:discretehalfbridge']} we consider the learning of stochastic half bridges in discrete state spaces with known forward dynamics and unknown backward dynamics which are inferred with a neural network. The learned dynamics then invert the known dynamics in time. The boundary conditions are discrete probability distributions denoted by $p_0(x)$ and $p_T(x)$.
  • Figure 3: In \ref{['cha:stochasticbridge']} we consider the learning of Schrödinger bridges in continuous state spaces with unknown forward and backward dynamics which are inferred with a neural network. Each of the learned dynamics is trained to reverse the opposite dynamics in time between the two probabilistic boundary conditions given by the two continuous distributions $p_0(x)$ and $p_T(x)$.
  • Figure 4: A figurative visualization of solving an ordinary differential equation.
  • Figure 5: A figurative visualization of solving an Ito drift-diffusion process. Integrating the deterministic dynamics $f(x, t)$ and the stochastic Wiener process $dW_t$ generates a conditional probability distribution $p_{t+ \Delta t}(x_{t+\Delta t} | x_t)$ over subsequent states $X_t$.
  • ...and 28 more figures

Theorems & Definitions (8)

  • Proposition 3.2.1: Reconstruction of Time-Reversible Dynamics
  • proof
  • Proposition 4.2.1
  • proof
  • Proposition 4.2.2
  • proof
  • Proposition 5.2.1
  • proof