Minimal-Dissipation Learning

• Minimal-dissipation learning is a framework that minimizes energetic, informational, and physical losses during state changes, using variational and thermodynamic principles.
• It employs optimization methods such as natural gradient flows and Rayleighian minimization to match theoretical lower bounds on dissipation.
• Applications span from neural network training to biochemical sensing, demonstrating enhanced stability and predictive performance in diverse systems.

Minimal-dissipation learning refers to the formulation, analysis, and implementation of learning processes—broadly encompassing neural, physical, and statistical systems—in which the energetic, informational, or physical cost associated with state changes, adaptation, or inference is minimized according to precise thermodynamic or information-theoretic criteria. The principle plays a central role in diverse domains, from biochemical sensing to energy-based machine learning, and is underpinned by variational and optimization frameworks that explicitly encode the dissipation, excess work, or entropy production incurred during finite-time, out-of-equilibrium transformations.

1. Formal Definitions and Core Principles

The concept of minimal-dissipation learning is rooted in both thermodynamic and information-theoretic formalisms. In thermodynamic systems, dissipation typically refers to the excess work or heat produced when a system is driven away from equilibrium—notably, the difference between actual work performed and the minimal work required for a quasistatic transformation. For learning systems and energy-based models, the bias in approximation or optimization can be mathematically linked to the excess thermodynamic work $\langle W_{\mathrm{ex}} \rangle$:

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where $L_{\mathrm{MLE}}$ is the ideal maximum-likelihood objective, and $L_\mathrm{approx}$ is what is achievable in finite time (Hnybida et al., 3 Oct 2025).

Minimal-dissipation protocols are obtained by minimizing quadratic forms (for weak driving, linear response) or more general energy- or information-based cost functionals over update schedules, transitions, or operator architectures. Notable examples include minimizing Rayleighians in Onsager-based operator learning (Chang et al., 10 Aug 2025), minimizing excess dissipation in fluid flows (Ruangkriengsin et al., 2022), and controlling learning rate schedules to match thermodynamic lower bounds (Hnybida et al., 3 Oct 2025).

2. Information-Theoretic Minimal-Dissipation Learning

Minimal dissipation in representation learning aims to produce statistics that are both sufficient for prediction and minimal in the sense of retaining no superfluous information (“leakage”). The MASS Learning framework (Cvitkovic et al., 2019) formalizes this by penalizing excess entropy in the representation via the conserved differential information (CDI):

$C(X, f(X)) = H(f(X)) - \mathbb{E}_X[\log J_f(X)]$

The full objective balances prediction loss with entropy and change-of-variables Jacobian regularization:

$$\mathcal{L}_\mathrm{MASS}(f) = H(Y|f(X)) + \beta H(f(X)) - \beta \mathbb{E}_X[\log J_f(X)]$$

Minimizing the CDI ensures that learned representations are “dissipation–minimal” in the information-theoretic sense, retaining only the essential structure necessary for prediction.

3. Thermodynamic Protocols and Optimal Transport

Thermodynamic approaches to minimal-dissipation learning leverage optimal control theory and stochastic process analysis to derive time-dependent protocols that minimize energetic losses or entropy production in finite-time learning or transformation regimes (Oikawa et al., 3 Mar 2025). The central insight is that for such finite–time processes, the excess dissipation is lower–bounded by geometric quantities (e.g., Wasserstein distance in optimal transport) and that optimal schedules may contain sharp transitions (“steps” or $\delta$–peaks) rather than being smooth (Bonança et al., 2018).

In energy-based learning systems driven by overdamped Langevin dynamics, the minimal excess work protocol can be realized by precise control of the learning rate $\eta(t)$, which synchronizes model update and sample relaxation so that the system tracks equilibrium as closely as possible:

$\eta(t) = \frac{1}{\tau - t + 1/\mu}$

where $\tau$ is the total protocol duration and $\mu$ is the mobility parameter. For general potentials, the optimal schedule takes the form of a natural gradient flow:

$$\dot{\theta} = - \frac{\beta}{\tau - t} g^{-1} \nabla_\theta L_\mathrm{MLE}$$

with $g$ the Fisher information metric (Hnybida et al., 3 Oct 2025).

4. Variational Principles and Operator Learning

Variational approaches express dissipative system evolution as the minimization of energy functionals or Rayleighians, often rooted in Onsager’s principle. In the DOOL framework (Chang et al., 10 Aug 2025), operator networks are trained directly through the minimization of the Rayleighian functional:

$$\mathcal{R}(u, \partial_t u, j) = \dot{\mathcal{E}}(u, \partial_t u) + \Phi(u, j)$$

subject to conservation laws (e.g., $\partial_t u + \nabla \cdot j = 0$), and minimizing this unsupervised loss yields dissipation–minimal predictions and evolution. For second–order systems, extensions use explicit damping factors in least action principles to encode dissipativity.

In dynamical system identification (e.g., learning neural Koopman operators (Xu et al., 8 Sep 2025)), dissipativity is enforced via linear matrix inequality (LMI) constraints and minimal perturbations, guaranteeing that the learned model obeys strict dissipative bounds—which are then theoretically extended back to the original nonlinear system even in the presence of noise and finite data.

5. Minimal-Dissipation in Physical and Biochemical Systems

Biological sensing modules and quantum systems highlight the fundamental trade-off between instantaneous response and predictive capacity: in the adiabatic (minimal dissipation) regime, systems maximize current-state information but perform poorly at nontrivial forecasting; in contrast, regimes of maximal dissipation (where response lags input) enable enhanced prediction of future environmental states, especially under non-Markovian or oscillatory driving (Becker et al., 2013). The relevant expressions for heat dissipation and predictive information are:

$$Q = 2\pi [1 - \sqrt{1 - \rho^2}] \nu_0 (\mu/\omega) / [1 + (\mu/\omega)^2]$$

$$I[n, \alpha_\tau] \approx (\nu_0 \rho^2 / 4) \left\{ \frac{ (\mu/\omega)^2 \cos(\omega\tau) - (\mu/\omega) \sin(\omega\tau) }{ 1 + (\mu/\omega)^2 } \right\}^2$$

Maximal predictive accuracy for long-range forecasting is reached at maximal dissipation (i.e., $\mu \approx \omega$), with the memory effect of dissipation resolving ambiguities in future state mapping.

In fluid dynamics, minimizing excess dissipation yields variational approximations and comparison principles for Stokes flows, which are foundational for both analytic techniques and machine learning–based field prediction (Ruangkriengsin et al., 2022).

6. Learning Frameworks, Guarantees, and Applications

Recent advances incorporate dissipative guarantees into deep learning model architectures by differentiably projecting arbitrary neural network–represented dynamics into dissipative subspaces (using the nonlinear Kalman–Yakubovich–Popov lemma) (Okamoto et al., 21 Aug 2024), or correcting learned models via minimal perturbation while maintaining original accuracy (Xu et al., 8 Sep 2025). These techniques assure stability, input–output stability, and strict energy conservation, with demonstrated robustness under dataset shifts and unpredictable inputs.

In unsupervised operator learning for dissipative PDEs, models such as DOOL achieve monotonic free energy decay and temporal extrapolation superior to supervised operator–learning approaches, driven by spatiotemporal decoupling and physically–motivated loss functions (Chang et al., 10 Aug 2025).

Experimental realizations of minimal dissipation have been achieved in stochastic thermodynamic systems, e.g., optically–trapped microparticles implementing protocols where the excess dissipation is set by the Wasserstein distance (Oikawa et al., 3 Mar 2025). Bounds relating speed, dissipation, and information erasure accuracy have been experimentally verified.

7. Theoretical Extensions and Outlook

Minimal-dissipation learning bridges thermodynamics, information theory, and modern AI by identifying and minimizing the physical and informational cost of adaptation in finite–time, often out–of–equilibrium regimes. Whether applied to inference, representation learning, physical system identification, or control, the core methodology involves:

• framing protocols or update rules as variational problems (quadratic form, action, or Rayleighian minimization);
• leveraging geometric quantities (relaxation functions, optimal transport distances, thermodynamic metrics) for schedule and architecture optimization;
• enforcing dissipativity via explicit constraints (LMIs, differentiable projections, Jacobian–regularized losses);
• linking energetic and predictive performance through explicit expressions and theoretical bounds.

Key implications include precise lower bounds on computational energy, guidance for robust and generalizable ML model design, and a unified language for stability, passivity, and thermodynamic efficiency across physical, biological, and computational systems. The balancing of minimal energy loss against predictive and organizational demands is a fundamental design principle that informs ongoing theoretical and applied research in both machine learning and statistical physics.