What if a neural network could turn hundreds of microscopic degrees of freedom into just three meaningful thermodynamic coordinates—and then use them to explain why some polymers stretch quickly while others get stuck? The modern scientific method requires theories to be hypothesized, tested on previously amassed data, compatible with basic scientific principles, and verifiable by experiments.

But there is no general algorithmic recipe for discovery in complex systems. As a result, intuitive equations mainly describe basic, often static equilibrium phenomena. Many dynamic, non-equilibrium phenomena in biology, soft-condensed matter and chemistry are instead described by very approximate empirical laws.

The advancement of artificial intelligence and machine learning creates the possibility of a data-driven solution to this challenge. S-OnsagerNet is an artificial intelligence platform that discovers an interpretable and closed thermodynamic description of an arbitrary stochastic dissipative dynamical system directly from microscopic trajectories.

The paper distinguishes unstructured approaches from structured approaches for understanding and predicting dynamical processes from data. Unstructured approaches parameterize dynamical equations with generic building blocks such as fixed polynomials, trainable feature maps or kernels, and fit parameters to observations.

Their generality comes at a cost of long-time predictive accuracy, stability and, most importantly, interpretability. Structured approaches let physical insights directly guide the design of model architectures, and this approach belongs to that category. Previous structured models include Hamiltonian or symplectic dynamics, Poisson systems and quasi-potentials.

So far, there has been no general structured approach for dissipative, non-equilibrium and noisy dynamics common in soft matter and biophysics. This methodology is based on the classical Onsager principle and is tailored to those problems.

Macroscopic thermodynamic descriptions are valuable because they provide insight; the ideal-gas law is a prototypical example for non-interacting gas systems. Such descriptions guide verification experiments and provide principled ways to manipulate macroscopic behavior, but constructing one for a general complex dynamical system is daunting.

The approach learns a macroscopic thermodynamic description by simultaneously constructing low-dimensional, partially interpretable closure coordinates and a time-evolution law on those coordinates. Unlike general artificial intelligence approaches, the platform intrinsically limits the search to physically relevant evolution laws through a neural-network architecture based on a generalized Onsager principle.

Figure one lays out the authors’ workflow from microscopic coordinates, X, to a target macroscopic coordinate, Z star, and learned closure coordinates, which together form the reduced state Z. The upper pathway illustrates the idea with the equilibrium, static ideal-gas law, while the lower pathway applies it to non-equilibrium polymer dynamics, where chain extension is modeled from complex configurations.

The figure matters because it shows how learned coordinates and a closed dynamical equation can turn microscopic system behavior into an interpretable macroscopic model. The method is demonstrated on polymer chains containing up to 900 degrees of freedom, condensed into a thermodynamic description involving only three macroscopic coordinates for computational and experimental data.

The learned energy landscape reveals stable and transition states, and can be viewed as a dynamic equation of state. Mastering this equation allows verification computational experiments, interpretation of the thermodynamic coordinates and control of the polymer stretching rate.

The complete microscopic description gives the coordinates of all components as a function of time. For an ideal gas, that means the positions and momenta of all molecules. A thermodynamic description replaces this expensive microscopic model with macroscopic coordinates whose dimensionality is much smaller than the microscopic coordinates.

Examples include the pressure of an ideal gas and the magnetization of a magnetic system. The thermodynamic approach links macroscopic coordinates to closure variables and external parameters through an equation of state. The generic approach builds custom thermodynamics for an arbitrary stochastic, dissipative dynamical system from data, starting with macroscopic coordinates whose dynamics should be modeled.

For polymer dynamics, the macroscopic coordinate can be the extension of the polymer chain; the method then learns closure variables and an evolution law on the combined thermodynamic coordinates. The generalized stochastic Onsager principle uses a symmetric positive semi-definite matrix for energy dissipation, an anti-symmetric matrix for conservative forces, and a diffusion matrix with white noise for thermal fluctuations.

The unknown functions are represented by trainable deep neural networks while preserving physical constraints, and another deep neural network generates closure coordinates using residual networks and approximate feature orthogonality through principal component analysis.

The first demonstration models the temporal evolution of polymer extension under elongational forces, where identical polymer chains show highly heterogeneous stretching dynamics. Because of complex interactions and stochastic behavior, identifying sufficient macroscopic descriptors and governing equations is challenging.

The machine learning method constructs closure coordinates with PCA-ResNet and governing equations on reduced coordinates with S-OnsagerNet, including drift and noise terms. The simulation uses a polymer chain of 300 coarse-grained beads connected by rigid rods, giving 900 degrees of freedom when inertial effects are ignored.

The projected chain length along the elongational axis is the macroscopic coordinate of interest, and the chain snapshots reveal highly heterogeneous extension dynamics. The goal is to model the stochastic evolution of that coordinate and understand the origin of its heterogeneity.

The approach adds two closure coordinates to the chain extension, producing a three-dimensional dynamical system with learned functions that governs stretching dynamics. Figure two first shows the simulation: a three-hundred-bead polymer chain with rigid bonds in planar elongational flow, and how its extension is measured along the flow axis.

The trajectory panels reveal that chains with similar initial extension can unfold through highly variable dynamics. The lower panels compare true and predicted extension statistics, including means, standard deviations, and unfolding-time probability distributions, showing why capturing both average behavior and trajectory-to-trajectory variability matters.

The chains start with similar extension lengths but subsequently stretch at vastly different rates. The true statistics, shown in black, are accurately predicted in red. The distributions of the time taken to reach a reference extension length are also successfully captured.

Only two closure coordinates are required to characterize the stochastic evolution of the extension length, and the method probes what those discovered coordinates mean. Because the neural network represents the nonlinear reduction functions as differentiable maps, automatic differentiation computes perturbations in directions of steepest changes in the closure coordinates.

Those conformational changes suggest physical interpretations: perturbations in the direction of the second closure coordinate tend to change the end-to-end distance along the elongational axis. The magnitude of the second closure coordinate correlates with end-to-end distance, while a similar analysis connects the third closure coordinate with foldedness along the flow direction.

Figure three gives the learned closure coordinates a physical interpretation. In panels a through c, automatic differentiation perturbs the polymer along the Z two direction, and the plotted data show that larger absolute Z two corresponds to a larger projected end-to-end distance; the example chains become less extended as absolute Z two decreases.

Panels d through f similarly connect Z three to foldedness, showing how these learned coordinates encode distinct geometric features of the chain rather than acting as abstract variables. The constructed potential can be interpreted as a generalized free energy, with local minima representing stable or metastable states and saddle points corresponding to transition states.

The figure shows two-dimensional projections of the three-dimensional free energy, and critical points are identified by solving the zero-gradient condition with the BFGS method. Two stable fixed points and two saddle points are found, and a PCA-ResNet decoder reconstructs polymer coordinates at those critical points.

Up to reflection symmetry, stable points correspond to fully stretched states, while saddle points correspond to completely folded states. A rapidly stretching polymer avoids the saddle and goes directly to a stable minimum, whereas a slowly stretching chain remains near the saddle's stable manifold before escaping through its unstable manifold.

Figure four maps the learned potential energy landscape across three pairs of latent coordinates, using both contour and surface views. Stable states appear as minima, while saddle points mark transition states; the plotted fast and slow trajectories show how nearby initial configurations can follow different unfolding paths, with the slow path lingering near a saddle.

The lower panels compare observed fluctuations around the stretched stable state with predicted contours, showing how the learned landscape captures local behavior. The slow trajectory escapes from the saddle's repulsive unstable manifold toward the stable minimum.

A fast and a slow unfolding trajectory are overlaid on the potential landscape to confirm this interpretation. Although the initial chain configurations are similar, the chains show different stretching behaviors that can be rationalized by the constructed potential.

Figure five shows a data-driven protocol for steering polymer unfolding. The authors turn the flow off near the folded-state saddle, allowing Brownian motion, then turn it back on when the reduced coordinates align with the saddle’s unstable manifold, as shown in panels a and b.

Panels c and e compare controlled and uncontrolled trajectories, while panels d and e illustrate a reversed protocol that slows unfolding, demonstrating how the inferred dynamics can be used to design polymer behavior. Qualitative predictions from the constructed thermodynamic description are confirmed by physical experiments, and the dynamical landscape supports fine-grained classification in simulation data and physical experiments.

Single-molecule experiments observe DNA stretching trajectories in a planar elongational field, including two samples that initially appear similar and cannot be visually distinguished by stretching behavior. When the samples are mapped into the learned thermodynamic coordinates, subtle differences in the second coordinate lead to different predicted stretching statistics.

This improves on human-level labeling, which can occur only much later in the dynamical evolution. The effective equation of state capturing correlations around the stretched state also applies to experimental data from the current experiments and previously available data.

Figure six connects the experimental setup to the model’s analysis of DNA stretching. The microfluidic cross-slot and electrodes trap and extend negatively charged DNA, while image sequences show two trajectories that look similar initially but are manually classified as dumbbell and folded states.

S-OnsagerNet maps these images into learned thermodynamic coordinates, reconstructs their configurations, and predicts distinct unfolding-time distributions and fluctuation patterns, showing how the learned landscape captures differences that are difficult to see directly.

To demonstrate general applicability, the method is applied to derive macroscopic dynamics of spatial epidemics. Using microscopic trajectories from the spatial SIR model, the thermodynamic model accurately describes the evolution of spatial averages of infective and susceptible individuals with one additional learned closure coordinate.

That coordinate can be interpreted as the spatial overlap of infective and susceptible individuals, which rationalizes the dynamical landscape where overlap determines the onset and outcome of disease spread. The search is constrained to macroscopic dynamics that conform to the generalized stochastic Onsager principle, limiting the method's ability to model systems such as chaotic systems that may not admit such a description.

The present model reduction and stability theory require thermal noise to be small compared with dissipative and conservative forces, and the theory needs expansion for highly stochastic cases. The dimension of closure variables is currently selected by trial and error, while accurate landscape construction requires trajectory data to sample stable and transition states sufficiently.

An adaptive or active learning algorithm could couple data sampling with S-OnsagerNet training, and future polymer studies could vary the stretching conditions. S-OnsagerNet learns a compact, physically constrained description of stochastic polymer dynamics, interprets its coordinates and energy landscape, and transfers qualitative predictions to DNA experiments.