Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization
Authors: Arthur N. Montanari, Francesco Bullo, Dmitry Krotov, Adilson E. Motter
Organizations: Center for Network Dynamics and the Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA · Center for Control, Dynamical Systems, and Computation and the Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA · Independent Researcher, Cambridge, MA 02141, USA · Center for Network Dynamics, the Department of Physics & Astronomy, the Northwestern Institute on Complex Systems, and the Department of Engineering Sciences & Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
Recent advances at the intersection of control theory, neuroscience, and machine learning have revealed novel mechanisms by which dynamical systems perform computation. These advances encompass a wide range of conceptual, mathematical, and computational ideas, with applications for model learning and training, memory retrieval, data-driven control, and optimization. This tutorial focuses on neuro-inspired approaches to computation that aim to improve scalability, robustness, and energy efficiency across such tasks, bridging the gap between artificial and biological systems. Particular emphasis is placed on energy-based dynamical models that encode information through gradient flows and energy landscapes. We begin by reviewing classical formulations, such as continuous-time Hopfield networks and Boltzmann machines, and then extend the framework to modern developments. These include dense associative memory models for high-capacity storage, oscillator-based networks for large-scale optimization, and proximal-descent dynamics for composite and constrained reconstruction. The tutorial demonstrates how control-theoretic principles can guide the design of next-generation neurocomputing systems, steering the discussion beyond conventional feedforward and backpropagation-based approaches to artificial intelligence.
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Fig. 1: Dynamical systems used in neurocomputation, machine learning, and unconventional computing that admit EDM formulations (the acronyms are defined throughout the text). In biological neural systems, computation is most commonly interpreted as emerging from spiking activity and complex recurrent circuitry. In contrast, machine learning models simplify this representation to layered neural architectures with feedforward signal propagation and static nonlinear activations, which facilitate training and inference. Unconventional computing, on the other hand, exploits the intrinsic physical dynamics of analog hardware to perform computation; many such systems, including electronic oscillators, photonic devices, resistor networks, and spintronics, exhibit oscillatory or nonlinear dynamical behavior. The EDM formalism presented in this tutorial provides a unifying framework to study and design neurocomputational models across these three domains.
Fig. 2: Continuous-time Hopfield neural network for associative memory. (a) Neural network, where nodes represent individual neurons xi and edges represent pairwise interactions Wij . The activity of each neuron encodes the pixel intensity of a grayscale image, forming a memory pattern ξ(μ) corresponding to the letter N. (b) Schematic of the energy landscape of a high-dimensional Hopfield network. Stored memory patterns ξ(μ) are encoded as local minima of E(x) , and inference occurs as the network state x(t) relaxes from an initial condition x(0) toward a nearby minimum through gradient descent. The energy landscape may contain undesired local minima, known as spurious attractors. (c) Energy landscape E(x) of a Hopfield network with N=2 neurons and K=2 patterns. The contour levels are shown in the (x1,x2) plane. (d) Hebbian learning adjusts the synaptic weights Wij so that the energy landscape can encode (store) the desired memory patterns as local minima. In this example, the energy landscape is initialized with random weights, yielding a monostable system. As the weights evolve according to Oja’s rule, the energy function encodes the desired patterns.
Fig. 3: Boltzmann machines for generative modeling and sampling-based inference. (a) Gibbs-Boltzmann distribution, plotted for increasing temperatures. States with higher energy are exponentially less likely to be sampled by the Boltzmann machine. (b) Evolution of the energy landscape via contrastive Hebbian learning. Blue and orange nodes represent states x sampled from the data distribution πdata and from the model distribution πθ , respectively. Through this contrastive learning, the model reshapes the landscape so that configurations consistent with the data occupy low-energy regions, while inconsistent configurations are pushed to higher energy. Thus, the learned model samples data-like states with higher probability.
Fig. 4: Oscillatory associative memory models. (a) Binary memory patterns are encoded as phase-locked configurations, which form stable equilibria due to the second-harmonic coupling in OAMs. (b) Capacity scaling for small-error (left) and error-free (right) retrieval. The curves are shown for Hopfield networks (assuming binary random, orthogonal, and sparse patterns), DAMs (with different functions F ), and OAMs (with different coupling strengths κ ). (c) Stability diagram of the OAM, showing the distribution of λmax(J(ξ(μ))) for stored binary memory patterns, single-bit perturbations of those patterns, and random binary configurations. For a suitable choice of κ , the desired memory patterns can be made stable while all other (undesired) configurations become unstable, thereby eliminating spurious attractors.
Fig. 5: Positive competitive network implementing sparse signal reconstruction via the proximal gradient dynamics. Each excitatory neuron i receives feedforward drive Θi⊤u proportional to its dictionary match, and is laterally inhibited by neuron j with strength −Θi⊤Θj≤0 (for nonnegative dictionaries). A global bias λ controls the sparsity level of the steady-state representation.
Fig. 6: E k -I network implementing winner-take-all competition. A set of k excitatory neurons project to a shared inhibitory interneuron, which in turn inhibits all excitatory neurons. Under the monostability condition wEE<1 and the functionality condition wIE≥1+wII , the network selects the neuron receiving the strongest input while suppressing all others.
A recurring pattern in neural computation is the reintroduction of dynamical and biological structure into models originally simplified for scalable optimization. Early feedforward networks reduced biological neurons to threshold or rate-like summation units, an abstraction compatible with global-gradient training at scale. Since then, forward computation has diversified: modern architectures carry recurrent state, retrieve from long contexts through attention and associative memory, compress histories through structured state-space dynamics, evolve in continuous time, settle to implicit equilibria, and communicate through sparse spikes. Training has diversified less. Scalable learning remains concentrated around backpropagation, backpropagation through time, adjoint methods, implicit differentiation, and surrogate-gradient variants. We call this asymmetry the forward-backward disconnect and develop a taxonomy spanning neural model families along three coupled axes: state-dynamics structure, credit-assignment mechanism, and biological grounding. Forward and learning grounding are treated separately, and the unit of analysis is the architecture-learning configuration rather than the architecture name alone. Across static, recurrent, attention-based, state-space, continuous-time, implicit, spiking, biologically plausible, and neuromorphic families, forward dynamics have diversified while the highest demonstrated scales remain concentrated in global or closely gradient-derived error-propagation mechanisms. Closing this disconnect requires better alignment among state dynamics, credit assignment, and computational substrate.
Hadi Al Mubasher, Mariette Awad
Department of Electrical and Computer Engineering, Maroun Semaan Faculty of Engineering and Architecture, American University of Beirut, Lebanon
Modern machine learning optimizes for accuracy without explicit treatment of internal computational cost, even though physical and biological systems operate under intrinsic energy constraints. We evaluate energy-aware learning across 2,203 experiments spanning vision, text, neuromorphic, and physiological datasets with 10 seeds per configuration and factorial statistical analysis. Three findings emerge. First, architecture alone explains negligible variance in accuracy (partial eta^2 = 0.001), while the architecture x dataset interaction is large (partial eta^2 = 0.44, p < 0.001), demonstrating that optimal architecture depends critically on task modality and rejecting the assumption of a universal best architecture. Second, a controlled lambda-sweep across lambda in {0, 1e-5, 1e-4, 1e-3, 1e-2} validates a single-parameter energy-regularized objective L = L_CE + lambda * E(theta, x): across this range, internal activation energy decreases by approximately three orders of magnitude relative to the unregularized lambda=0 baseline, with negligible accuracy change (<0.5 percentage points) on both MNIST and Fashion-MNIST. Third, energy-first architectures inspired by an action-principle framework yield 5-33% within-modality training-efficiency gains over conventional baselines. These results emerge from a research program that interprets learning through a structural correspondence between the action functional in classical mechanics, free energy in statistical physics, and KL-regularized objectives in variational inference. We frame this correspondence as a design hypothesis, not a derivation.
Martin G. Frasch
Institute on Human Development and Disability, University of Washington, Seattle, WA 98195, USA · Health Stream Analytics, LLC, Seattle, WA, USA
Learning and decision-making in animals are often modeled as Bayesian processes, where sensory evidence is integrated with prior beliefs to guide behavior in the face of uncertainty. But what are the inherent neural dynamics that give rise to this ability, and how could they be replicated in computing systems? This abstract discusses a biologically grounded framework in which noisy neural and synaptic dynamics perform inference and learning via stochastic sampling from an internal energy function, capturing uncertainty over latent states and model parameters through neural and synaptic variability, respectively. This enables approaches such as predictive coding networks to account for epistemic uncertainty via Markov chain Monte Carlo sampling. Drawing a parallel between intrinsic noise in biological systems and electrical noise in emerging probabilistic analogue memory technologies, we highlight how analogue in-memory computing hardware naturally emerges as the solution for massively scalable and energy-efficient probabilistic inference.