Latent Thermodynamic Flows (LaTF)

• LaTF is a unified framework that models and generates equilibrium distributions of complex thermodynamic systems in low-dimensional latent spaces.
• It integrates state-predictive encoding with normalizing flows to accurately capture metastable states and reconstruct free energy landscapes.
• The approach enables data-efficient, temperature-dependent sampling and analysis, offering actionable insights for thermodynamic system exploration.

Latent Thermodynamic Flows (LaTF) describe a unified framework for modeling, representing, and generating equilibrium distributions of complex thermodynamic systems in low-dimensional latent spaces. This framework tightly couples information-theoretic representation learning with generative modeling to explicitly encode temperature-dependent and metastable behaviors, enabling both interpretability and accurate, data-driven generation of structural ensembles across varied thermodynamic conditions (2507.03174).

1. Unified Framework Architecture

Latent Thermodynamic Flows operate as an end-to-end system. The starting point is a high-dimensional molecular descriptor (e.g., molecular coordinates, dihedrals, or features from molecular dynamics trajectories), denoted $X_t$, which is processed by an encoder network to produce a low-dimensional representation—commonly called the information bottleneck (IB) or collective variables (CVs). These latent CVs are designed to encapsulate the system’s slow, thermodynamically relevant degrees of freedom.

Simultaneously, a state-predictive decoder predicts the likely metastable state after a lag time, $\Delta t$, using this latent representation. Crucially, the encoding, decoding, and the training of a normalizing flow (NF) generative model in the latent space are all optimized jointly:

• The encoder maps $X_t$ to $z$ (IB space).
• The decoder $q_\theta(y_{t+\Delta t} | z)$ predicts future state membership.
• A normalizing flow $\mathcal{F}_\theta$ transforms a latent noise sample $u$ to the expressive IB space, enabling flexible prior matching, sampling, and evaluation of density via the change-of-variable formula.

The total loss for joint training is:

$$\mathcal{L}_\text{LaTF} = -\int du~p_\theta(u|X_t) \left[\log q_\theta(y_{t+\Delta t}|\mathcal{F}_\theta(u)) + \beta \log r(\mathcal{F}_\theta(u)) + \beta \log \left| \det \frac{\partial \mathcal{F}_\theta(u)}{\partial u} \right| \right]$$

where $\beta$ tunes regularization, $r(\cdot)$ is the prior, and $p_\theta(u|X_t)$ is the stochastic encoder distribution (2507.03174).

2. State Predictive Information Bottleneck (SPIB) and Metastable State Discovery

The SPIB principle, fundamental to LaTF, focuses the representation learning on extracting only the “useful” information in $X_t$ for predicting the system’s future state at $t+\Delta t$. Architecturally, this involves:

• Learning an encoding of $X_t$ to $z$ (IB space) that is minimal and discards irrelevant fast modes, yet maximally predictive of coarse-grained state transitions.
• A cross-entropy loss for the future state prediction, regularized by a KL-divergence that constrains the IB distribution to a known prior, ensuring the encoded space remains well-structured.

SPIB automatically differentiates slow (“collective”) from fast (“noise”) degrees of freedom by merging accidentally short-lived states and iteratively relabeling metastable state assignments, promoting representations that have kinetic and thermodynamic relevance to the system's transitions.

3. Generative Modeling with Normalizing Flows

The normalizing flow component in LaTF consists of a stack of invertible transformations (e.g., RealNVP layers) that map simple reference distributions (typically a standard or modified Gaussian) into the complex, multimodal equilibrium distribution of the system within the latent IB space.

Mathematically, the mapping $u \mapsto z = \mathcal{F}_\theta(u)$ is paired with an exactly computable Jacobian determinant, which (via change-of-variable formula) gives the latent density:

$$p(z) = p(u) \left| \det \frac{\partial \mathcal{F}_\theta^{-1}(z)}{\partial z} \right|$$

The NF aids both discriminative tasks (e.g., better CVs for state separation) and generative tasks (sampling equilibrium ensembles or interpolating transition pathways).

4. Collective Variables, Free Energy Landscapes, and Temperature Dependence

LaTF yields a CV space where projections of molecular conformations enable:

• Clear classification and visualization of metastable states,
• Construction of free energy surfaces (FES) for interpretation,
• Efficient sampling and interpolation of system conformational ensembles.

A key innovation in LaTF is the introduction of a temperature-steerable prior for the NF. Instead of a regular Gaussian prior, an exponentially-tilted Gaussian is used:

$$r_T(z, \tau) \propto \exp(\tau \|z\|) \exp\left( -\frac{(\|z\| - T\tau)^2}{2T} \right)$$

This distribution allows the latent model to modulate both the width (i.e., variance/entropy) and the location of high-density regions as a function of temperature, thereby accommodating entropic broadening and temperature-dependent shift of the equilibrium ensemble (2507.03174).

5. Applications and Benchmarking

LaTF has been validated across a range of systems, demonstrating performance on both toy and complex molecular examples:

• 2D three-hole potential: LaTF recovers the original free energy landscape, with generated samples well matching simulation data.
• Chignolin protein: The method identifies folded, misfolded, and unfolded states in the CV space, and reveals transition pathways that are consistent with transition path theory predictions.
• Lennard-Jones 7-cluster: Competing energetic and entropic states are discovered, and FES is extended accurately across temperatures, despite limited training data.
• RNA tetraloop (RYYGG motif): Trained on simulations at only two temperatures (300 K and 400 K), LaTF reconstructs a six-state FES and predicts melting behavior as a function of temperature, with results consistent with experiment and extensive replica-exchange simulations.

Quantitative metrics (e.g., KL divergence between generated and empirical latent densities; generalized matrix Rayleigh quotient (GMRQ) scores) show LaTF outperforms vanilla SPIB or other two-stage approaches in both generative and classification fidelity.

6. Physical and Practical Significance

The LaTF approach provides:

• Data-efficient sampling: Accurate modeling of temperature-dependent ensembles and free energy surfaces with limited training sets, thanks to temperature steerability and shared latent structure.
• Unified workflow for analysis and generation: Direct use of learned CVs for metastable state assignment, physical interpretation, and unbiased equilibrium sampling.
• Transferability: The latent representation learned can extrapolate to new, unobserved conditions within physical limits, capturing melting transitions and shifts in state populations.

A plausible implication is that LaTF’s architecture enables integration with enhanced sampling and replica-exchange protocols to further boost sampling in challenging regimes. The latent-dynamical space is structured for physical interpretability, offering a bridge between information-theoretic and thermodynamic modeling for complex systems.

7. Extensions and Future Directions

The authors note that the LaTF methodology could be generalized beyond its present form by:

• Incorporating alternative representation learning modules (e.g., time-lagged autoencoders instead of SPIB),
• Integrating diffusion-based generative models to potentially increase expressivity,
• Making the prior’s tilting factor, $\tau$, a learnable function of temperature or conditioning variables to further refine temperature extrapolation,
• Combining with enhanced sampling datasets to leverage variance in out-of-equilibrium trajectories.

Potential theoretical advances could provide guarantees on the invariance, sufficiency, or robustness of the learned latent representation under varying thermodynamic conditions.

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Latent Thermodynamic Flows (LaTF) thus constitute a comprehensive, physically grounded framework for reducing complex many-body thermodynamic behavior into interpretable, generative latent models. Their joint optimization of state-predictive encoding and flow-based generation enables not only efficient classification and pathway discovery but also temperature- and condition-sensitive ensemble generation—all from limited data (2507.03174).