Energy-Based Models

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Overview

Energy-based models represent how compatible a data point or configuration is with a task: lower energy generally means greater plausibility, without requiring probabilities to be normalized explicitly. They are trained with methods such as contrastive divergence, noise-contrastive estimation, and score or energy matching, then used for inference and generation with Langevin MCMC, tempering, or diffusion and flow-based samplers. Work spans Boltzmann machines and Hopfield networks as well as image and molecule generation, structured prediction, robotics, and scientific modeling. A central challenge is making sampling and training efficient and reliable; in particular, equilibrium guarantees may not reflect finite-time outputs, and lower energy does not always mean better predictions or reasoning.

Latest papers

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91 papers9 papers in the last four weeks, up 125% on the four weeks before. 0.1% of all new papers.

May 7, 2026

Energy Generative Modeling: A Lyapunov-based Energy Matching Perspective

Generative models based on static scalar energy functions represent an emerging paradigm in which a single time independent potential drives sample generation through its gradient field, eliminating the need for time conditioning entirely. We unify the training and sampling phases of this paradigm, conventionally treated as separate procedures, within a single framework: density transport on the Wasserstein space, cast as a nonlinear control problem in which the Kullback Leibler (KL) divergence serves as a Lyapunov function. Training and sampling are then two instances of this same master dynamics, differing only in initial condition. Within this autonomous framework we develop two analytic results. First, since the Lyapunov certificate is asymptotic, we derive a finite step stopping criterion for Langevin sampling and prove that no Lyapunov certificate exists for the deterministic gradient flow on the same energy landscape. Second, the reformulation brings the toolkit of nonlinear control theory to bear on static scalar energy generative modeling, that is, we show that additive composition of trained scalar energies retains an explicit Gibbs invariant measure and inherits the closed-loop Lyapunov certificate. Beyond these immediate results, this reformulation bridges static scalar energy generative models with the full toolkit of nonlinear control theory, opening the door to barrier functions for constrained generation and contraction metrics for accelerated sampling. Experiments on synthetic distributions validate the theoretical predictions.

Yixuan Wang, Wenqian Xue, Warren E. Dixon

Department of Mechanical and Aerospace Engineering University of Florida

Energy-Based Models
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May 5, 2026

Flow Sampling: Learning to Sample from Unnormalized Densities via Denoising Conditional Processes

Sampling from unnormalized densities is analogous to the generative modeling problem, but the target distribution is defined by a known energy function instead of data samples. Because evaluating the energy function is often costly, a primary challenge is to learn an efficient sampler. We introduce Flow Sampling, a framework built on diffusion models and flow matching for the data-free setting. Our training objective is conditioned on a noise sample and regresses onto a denoising diffusion drift constructed from the energy function. In contrast, diffusion models' objective is conditioned on a data sample and regresses onto a noising diffusion drift. We utilize the interpolant process to minimize the number of energy function evaluations during training, resulting in an efficient and scalable method for sampling unnormalized densities. Furthermore, our formulation naturally extends to Riemannian manifolds, enabling diffusion-based sampling in geometries beyond Euclidean space. We derive a closed-form formula for the conditional drift on constant curvature manifolds, including hyperspheres and hyperbolic spaces. We evaluate Flow Sampling on synthetic energy benchmarks, small peptides, large-scale amortized molecular conformer generation, and distributions supported on the sphere, demonstrating strong empirical performance.

Aaron Havens, Brian Karrer, Neta Shaul

FAIR at Meta · Weizmann Institute of Science

Sampling from Unnormalized DistributionsDiffusion SamplingEnergy-Based ModelsDiffusion Models
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May 1, 2026

Learning Multimodal Energy-Based Model with Multimodal Variational Auto-Encoder via MCMC Revision

Energy-based models (EBMs) are a flexible class of deep generative models and are well-suited to capture complex dependencies in multimodal data. However, learning multimodal EBM by maximum likelihood requires Markov Chain Monte Carlo (MCMC) sampling in the joint data space, where noise-initialized Langevin dynamics often mixes poorly and fails to discover coherent inter-modal relationships. Multimodal VAEs have made progress in capturing such inter-modal dependencies by introducing a shared latent generator and a joint inference model. However, both the shared latent generator and joint inference model are parameterized as unimodal Gaussian (or Laplace), which severely limits their ability to approximate the complex structure induced by multimodal data. In this work, we study the learning problem of the multimodal EBM, shared latent generator, and joint inference model. We present a learning framework that effectively interweaves their MLE updates with corresponding MCMC refinements in both the data and latent spaces. Specifically, the generator is learned to produce coherent multimodal samples that serve as strong initial states for EBM sampling, while the inference model is learned to provide informative latent initializations for generator posterior sampling. Together, these two models serve as complementary models that enable effective EBM sampling and learning, yielding realistic and coherent multimodal EBM samples. Extensive experiments demonstrate superior performance for multimodal synthesis quality and coherence compared to various baselines. We conduct various analyses and ablation studies to validate the effectiveness and scalability of the proposed multimodal framework.

Jiali Cui, Zhiqiang Lao, Heather Yu

Basking Ridge, New Jersey, USA Futurewei Technologies Inc · Basking Ridge, New Jersey, USA2026 Futurewei Technologies Inc

Variational AutoencodersMarkov Chain Monte CarloEnergy-Based ModelsMultimodal Generation
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May 1, 2026

Energy-Based Constraint Networks: Learning Structural Coherence Across Modalities

We introduce energy-based constraint networks -- a modality-agnostic architecture that learns structural coherence from contrastive pairs. The system processes frozen encoder embeddings through a state-space model with dual-head attention, producing a scalar energy measuring structural consistency alongside per-position energy scores that localize violations. Multiple independently trained branches detect different violation types and compose at inference without interference. We demonstrate the framework in two domains. In text, the system achieves 93.4% accuracy on trained corruption types and 87.2% on 9 unseen types, using frozen BERT and 7.4M trainable parameters. In vision, the same architecture achieves competitive deepfake detection: 0.959 AUC on FaceForensics++ Deepfakes and 0.870 on Celeb-DF without any Celeb-DF training data, using frozen DINOv2 and 3.6M parameters per branch. The framework supports flexible training: branches learn from designer-specified corruptions, real-world paired data, or both. Composable branches require representation compatibility -- a finding validated through extensive experimentation where five incompatible approaches failed before the compatible one succeeded. The architecture is encoder-agnostic and domain-agnostic: changing the domain requires only new corruption strategies; changing the encoder requires only a new input projection layer. To our knowledge, this is the first architecture to learn within-modality structural coherence as an explicit energy landscape with per-position decomposition, and to demonstrate that the same architecture transfers across modalities via corruption respecification alone.

Chirag Shinde

Independent Researcher

Energy-Based ModelsAnomaly Detection
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May 1, 2026

Recovering Hidden Reward in Diffusion-Based Policies

This paper introduces EnergyFlow, a framework that unifies generative action modeling with inverse reinforcement learning by parameterizing a scalar energy function whose gradient is the denoising field. We establish that under maximum-entropy optimality, the score function learned via denoising score matching recovers the gradient of the expert's soft Q-function, enabling reward extraction without adversarial training. Formally, we prove that constraining the learned field to be conservative reduces hypothesis complexity and tightens out-of-distribution generalization bounds. We further characterize the identifiability of recovered rewards and bound how score estimation errors propagate to action preferences. Empirically, EnergyFlow achieves state-of-the-art imitation performance on various manipulation tasks while providing an effective reward signal for downstream reinforcement learning that outperforms both adversarial IRL methods and likelihood-based alternatives. These results show that the structural constraints required for valid reward extraction simultaneously serve as beneficial inductive biases for policy generalization. The code is available at https://github.com/sotaagi/EnergyFlow.

Yanbiao Ji, Qiuchang Li, Yuting Hu +7

Shanghai Jiao Tong University · KOKONI 3D

Reinforcement LearningDiffusion PolicyEnergy-Based Models
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Apr 26, 2026

ESIA: An Energy-Based Spatiotemporal Interaction-Aware Framework for Pedestrian Intention Prediction

Recent advances in autonomous driving have motivated research on pedestrian intention prediction, which aims to infer future crossing decisions and actions by modeling temporal dynamics, social interactions, and environmental context. However, existing studies remain constrained by oversimplified multi-agent interaction patterns, opaque reasoning logic, and a lack of global consistency in behavioral predictions, which compromise both robustness and interpretability. In this work, we propose ESIA (Energy-based Spatiotemporal Interaction-Aware framework), a novel Conditional Random Field (CRF)-based paradigm. We cast the intention prediction task as a structured prediction problem over a unified graph-based representation, treating pedestrians and the environment as spatiotemporal nodes. To characterize their distinct roles, we assign unary potentials to nodes to capture individual intentions, and pairwise potentials to edges to encode social and environmental interactions. These potentials are integrated into a unified global energy function to ensure scene-level consistency across behavioral predictions. To further constrain inference without ground-truth supervision, we introduce structural consistency terms to penalize logical contradictions. This optimization is efficiently solved via a novel Unary-Seeded Simulated Annealing (U-SSA) algorithm, which leverages high-confidence unary priors to rapidly converge to a high-quality solution. Extensive experiments on standard benchmarks demonstrate that ESIA achieves state-of-the-art performance with improved interpretability over existing methods.

Yanping Wu, Meiting Dang, Lin Wu +3

James Watt School of Engineering, University of Glasgow, Glasgow, G12 8QQ, United Kingdom · School of Computing Science, University of Glasgow, Glasgow, G12 8QQ, United Kingdom

Pedestrian Intention PredictionPedestrian Behavior ModelingConditional Random FieldsAutonomous DrivingEnergy-Based Models
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Apr 22, 2026

Energy-Based Open-Set Active Learning for Object Classification

Active learning (AL) has emerged as a crucial methodology for minimizing labeling costs in deep learning by selecting the most valuable samples from a pool of unlabeled data for annotation. Traditional AL operates under a closed-set assumption, where all classes in the dataset are known and consistent. However, real-world scenarios often present open-set conditions in which unlabeled data contains both known and unknown classes. In such environments, standard AL techniques struggle. They can mistakenly query samples from unknown categories, leading to inefficient use of annotation budgets. In this paper, we propose a novel dual-stage energy-based framework for open-set AL. Our method employs two specialized energy-based models (EBMs). The first, an energy-based known/unknown separator, filters out samples likely to belong to unknown classes. The second, an energy-based sample scorer, assesses the informativeness of the filtered known samples. Using the energy landscape, our models distinguish between data points from known and unknown classes in the unlabeled pool by assigning lower energy to known samples and higher energy to unknown samples, ensuring that only samples from classes of interest are selected for labeling. By integrating these components, our approach ensures efficient and targeted sample selection, maximizing learning impact in each iteration. Experiments on 2D (CIFAR-10, CIFAR-100, TinyImageNet) and 3D (ModelNet40) object classification benchmarks demonstrates that our framework outperforms existing approaches, achieving superior annotation efficiency and classification performance in open-set environments.

Zongyao Lyu, William J. Beksi

The University of Texas at Arlington, Arlington TX 76019, USA

Active LearningOpen-Set RecognitionEnergy-Based ModelsImage Classification
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Apr 19, 2026

StrEBM: A Structured Latent Energy-Based Model for Blind Source Separation

This paper proposes StrEBM, a structured latent energy-based model for source-wise structured representation learning. The framework is motivated by a broader goal of promoting identifiable and decoupled latent organization by assigning different latent dimensions their own learnable structural biases, rather than constraining the entire latent representation with a single shared energy. In this sense, blind source separation is adopted here as a concrete and verifiable testbed, through which the evolution of latent dimensions toward distinct underlying components can be directly examined. In the proposed framework, latent trajectories are optimized directly together with an observation-generation map and source-wise structural parameters. Each latent dimension is associated with its own energy-based formulation, allowing different latent components to gradually evolve toward distinct source-like roles during training. In the present study, this source-wise energy design is instantiated using Gaussian-process-inspired energies with learnable length-scales, but the framework itself is not restricted to Gaussian processes and is intended as a more general structured latent EBM formulation. Experiments on synthetic multichannel signals under linear and nonlinear mixing settings show that the proposed model can recover source components effectively, providing an initial empirical validation of the framework. At the same time, the study reveals important optimization characteristics, including slow late-stage convergence and reduced stability under nonlinear observation mappings. These findings not only clarify the practical behavior of the current GP-based instantiation, but also establish a basis for future investigation of richer source-wise energy families and more robust nonlinear optimization strategies.

Yuan-Hao Wei

Disentangled Representation LearningLatent Variable ModelsEnergy-Based Models
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Apr 8, 2026

Energy-based Tissue Manifolds for Longitudinal Multiparametric MRI Analysis

We propose a geometric framework for longitudinal multi-parametric MRI analysis based on patient-specific energy modelling in sequence space. Rather than operating on images with spatial networks, each voxel is represented by its multi-sequence intensity vector (T1T1, T1cT1c, T2T2, FLAIR, ADC), and a compact implicit neural representation is trained via denoising score matching to learn an energy function Eθ(u)E_θ(\mathbf{u}) over Rd\mathbb{R}^d from a single baseline scan. The learned energy landscape provides a differential-geometric description of tissue regimes without segmentation labels. Local minima define tissue basins, gradient magnitude reflects proximity to regime boundaries, and Laplacian curvature characterises local constraint structure. Importantly, this baseline energy manifold is treated as a fixed geometric reference: it encodes the set of contrast combinations observed at diagnosis and is not retrained at follow-up. Longitudinal assessment is therefore formulated as evaluation of subsequent scans relative to this baseline geometry. Rather than comparing anatomical segmentations, we analyse how the distribution of MRI sequence vectors evolves under the baseline energy function. In a paediatric case with later recurrence, follow-up scans show progressive deviation in energy and directional displacement in sequence space toward the baseline tumour-associated regime before clear radiological reappearance. In a case with stable disease, voxel distributions remain confined to established low-energy basins without systematic drift. The presented cases serve as proof-of-concept that patient-specific energy manifolds can function as geometric reference systems for longitudinal mpMRI analysis without explicit segmentation or supervised classification, providing a foundation for further investigation of manifold-based tissue-at-risk tracking in neuro-oncology.

Kartikay Tehlan, Lukas Förner, Sina Wendrich +5

Dept. of diagnostic and interventional Radiology and Neuroradiology, University Hospital Augsburg, Germany · Digital Medicine, University Hospital Augsburg, Germany · Chair for Computer Aided Medical Procedures and Augmented Reality, Technical University of Munich, Germany +3

Longitudinal Medical Image AnalysisMagnetic Resonance ImagingEnergy-Based ModelsHealthcareManifold Learning
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Apr 6, 2026

Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization

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.

Arthur N. Montanari, Francesco Bullo, Dmitry Krotov +1

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 +1

Energy-Based ModelsNeural Network OptimizationNeuromorphic ComputingAssociative MemoryHopfield Networks
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Apr 2, 2026

SCALE: Semantic- and Confidence-Aware Conditional Variational Autoencoder for Zero-shot Skeleton-based Action Recognition

Zero-shot skeleton-based action recognition (ZSAR) aims to recognize action classes without any training skeletons from those classes, relying instead on auxiliary semantics from text. Existing approaches frequently depend on explicit skeleton-text alignment, which can be brittle when action names underspecify fine-grained dynamics and when unseen classes are semantically confusable. We propose SCALE, a lightweight and deterministic Semantic- and Confidence-Aware Listwise Energy-based framework that formulates ZSAR as class-conditional energy ranking. SCALE builds a text-conditioned Conditional Variational Autoencoder where frozen text representations parameterize both the latent prior and the decoder, enabling likelihood-based evaluation for unseen classes without generating samples at test time. To separate competing hypotheses, we introduce a semantic- and confidence-aware listwise energy loss that emphasizes semantically similar hard negatives and incorporates posterior uncertainty to adapt decision margins and reweight ambiguous training instances. Additionally, we utilize a latent prototype contrast objective to align posterior means with text-derived latent prototypes, improving semantic organization and class separability without direct feature matching. Experiments on NTU-60 and NTU-120 datasets show that SCALE consistently improves over prior VAE- and alignment-based baselines while remaining competitive with diffusion-based methods.

Soroush Oraki, Feng Ding, Jie Liang

School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada · School of Information Science and Technology, Eastern Institute of Technology, Ningbo, China

Skeleton-Based Action RecognitionZero-Shot Action RecognitionZero-Shot LearningEnergy-Based ModelsConditional Variational Autoencoder
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Mar 31, 2026

Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics

Energy-based models (EBMs) implement inference as gradient descent on a learned Lyapunov function, yielding interpretable, structure-preserving alternatives to black-box neural ODEs and aligning naturally with physical AI. Yet their use in system identification remains limited, and existing architectures lack formal stability guarantees that globally preclude unstable modes. We address this gap by introducing an EBM framework for system identification with stable, dissipative, absorbing invariant dynamics. Unlike classical global Lyapunov stability, absorbing invariance expands the class of stability-preserving architectures, enabling more flexible and expressive EBMs. We extend EBM theory to nonsmooth activations by establishing negative energy dissipation via Clarke derivatives and deriving new conditions for radial unboundedness, exposing a stability-expressivity tradeoff in standard EBMs. To overcome this, we introduce a hybrid architecture with a dynamical visible layer and static hidden layers, prove absorbing invariance under mild assumptions, and show that these guarantees extend to port-Hamiltonian EBMs. Experiments on metric-deformed multi-well and ring systems validate the approach, showcasing how our hybrid EBM architecture combines expressivity with sound and provable safety guarantees by design.

Simone Betteti, Luca Laurenti

RIAS Lab, The Italian Institute of Artificial Intelligence for Industry, Turin, 10129, IT · Delft Center for Systems and Control, TU Delft, Delft, 2600 AA, The Netherlands

Energy-Based ModelsDynamical SystemsNonlinear System IdentificationLyapunov StabilityPort-Hamiltonian Systems
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Mar 30, 2026

Reasoning as Energy Minimization over Structured Latent Trajectories

Single-shot neural decoders commit to answers without iterative refinement, while chain-of-thought methods introduce discrete intermediate steps but lack a scalar measure of reasoning progress. We propose Energy-Based Reasoning via Structured Latent Planning (EBRM), which models reasoning as gradient-based optimization of a multi-step latent trajectory z1:Tz_{1:T} under a learned energy function E(hx,z)E(h_x, z). The energy decomposes into per-step compatibility, transition consistency, and trajectory smoothness terms. Training combines supervised encoder-decoder learning with contrastive energy shaping using hard negatives, while inference performs gradient descent or Langevin dynamics over zz and decodes from zTz_T. We identify a critical failure mode: on CNF logic satisfaction, latent planning reduces accuracy from ≈95%\approx 95\% to ≈56%\approx 56\%. This degradation arises from a distribution mismatch, where the decoder is trained on encoder outputs hxh_x but evaluated on planner outputs zTz_T that drift into unseen latent regions. We analyze this behavior through per-step decoding, latent drift tracking, and gradient decomposition. To address it, we propose dual-path decoder training and latent anchoring. We further introduce a six-part ablation protocol covering component contributions, trajectory length, planner dynamics, initialization, decoder training distribution, and anchor weight. Experiments on three synthetic tasks show that energy decreases monotonically and induces structured latent trajectories on graph and logic tasks, while remaining flat on arithmetic (r=0.073r = 0.073), indicating a negative result. Code is available at https://github.com/dkjo8/ebr-via-structured-latent-planning.

David K. Johansson

Polished Snow Inc.

Latent-Space PlanningContinuous Latent ReasoningEnergy-Based ModelsLogical ReasoningDistribution Shift
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Mar 24, 2026

Graph Energy Matching: Transport-Aligned Energy-Based Modeling for Graph Generation

Generative modeling of discrete data, such as graphs, underpins many scientific and industrial applications, including molecular discovery and materials design. In these domains, probabilistic inference is particularly valuable, as it enables composable generation and principled incorporation of desired constraints, such as structural or functional properties. Energy-based models naturally support this goal by capturing relative likelihoods and enabling composable inference by directly enforcing constraints during inference. However, discrete energy-based models typically struggle with efficient and high-quality sampling, as off-support regions often contain spurious local minima, trapping samplers and causing training instabilities, resulting in a fidelity gap compared to discrete diffusion models. To address this gap, we introduce Graph Energy Matching (GEM), a discrete generative framework inspired by the Jordan-Kinderlehrer-Otto (JKO) transport-map optimization perspective. GEM learns a permutation-invariant potential energy that simultaneously guides discrete transport from noise toward high-likelihood graph regions and refines samples within these regions. We further introduce a sampling protocol leveraging an energy-based switching strategy, seamlessly bridging rapid, gradient-guided transport and a local mixing regime for effective exploration. On molecular graph benchmarks, GEM matches or surpasses strong discrete diffusion baselines on most reported metrics. Beyond improving generation quality, GEM's relative likelihood modeling enables targeted exploration, facilitating compositional generation, property-constrained sampling, and interpolation between graphs. Project page: https://michalbalcerak.ai/graph-energy-matching/.

Michal Balcerak, Suprosanna Shit, Chinmay Prabhakar +4

University of Zurich · Harvard University · Kempner Institute

Generative ModelingEnergy-Based ModelsMolecular Graph GenerationConstrained Generative Modeling
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Mar 13, 2026

State Algebra for Probabilistic Logic

This paper presents a Probabilistic State Algebra as an extension of deterministic propositional logic, providing a computational framework for constructing Markov Random Fields (MRFs) through pure linear algebra. By mapping logical states to real-valued coordinates interpreted as energy potentials, we define an energy-based model where global probability distributions emerge from coordinate-wise Hadamard products. This approach bypasses the traditional reliance on graph-traversal algorithms and compiled circuits, utilising tt-objects and wildcards to embed logical reduction natively within matrix operations. We demonstrate that this algebra constructs formal Gibbs distributions, offering a rigorous mathematical link between symbolic constraints and statistical inference. A central application of this framework is the development of Probabilistic Rule Models (PRMs), which are uniquely capable of incorporating both probabilistic associations and deterministic logical constraints simultaneously. These models are designed to be inherently interpretable, supporting a human-in-the-loop approach to decisioning in high-stakes environments such as healthcare and finance. By representing decision logic as a modular summation of rules within a vector space, the framework ensures that complex probabilistic systems remain auditable and maintainable without compromising the rigour of the underlying configuration space.

Dmitry Lesnik, Tobias Schäfer

Stratyfy Inc., New York, NY, USA · Department of Mathematics, College of Staten Island, Staten Island, NY, USA & Physics Program, CUNY Graduate Center, NY, USA

Probabilistic LogicMarkov Random FieldsEnergy-Based Models
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Mar 12, 2026

EB-RANSAC: Random Sample Consensus based on Energy-Based Model

Random sample consensus (RANSAC), which is based on a repetitive sampling from a given dataset, is one of the most popular robust estimation methods. In this study, an energy-based model (EBM) for robust estimation that has a similar scheme to RANSAC, energy-based RANSAC (EB-RANSAC), is proposed. EB-RANSAC is applicable to a wide range of estimation problems similar to RANSAC. However, unlike RANSAC, EB-RANSAC does not require a troublesome sampling procedure and has only one hyperparameter. The effectiveness of EB-RANSAC is numerically demonstrated in two applications: a linear regression and maximum likelihood estimation.

Muneki Yasuda, Nao Watanabe, Kaiji Sekimoto

Graduate School of Science and Engineering, Yamagata University · TSCSK Corporation

Robust EstimationRobust StatisticsEnergy-Based Models
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Mar 9, 2026

EnergyAction: Unimanual to Bimanual Composition with Energy-Based Models

Recent advances in unimanual manipulation policies have achieved remarkable success across diverse robotic tasks through abundant training data and well-established model architectures. However, extending these capabilities to bimanual manipulation remains challenging due to the lack of bimanual demonstration data and the complexity of coordinating dual-arm actions. Existing approaches either rely on extensive bimanual datasets or fail to effectively leverage pre-trained unimanual policies. To address this limitation, we propose EnergyAction, a novel framework that compositionally transfers unimanual manipulation policies to bimanual tasks through the Energy-Based Models (EBMs). Specifically, our method incorporates three key innovations. First, we model individual unimanual policies as EBMs and leverage their compositional properties to compose left and right arm actions, enabling the fusion of unimanual policies into a bimanual policy. Second, we introduce an energy-based temporal-spatial coordination mechanism through energy constraints, ensuring the generated bimanual actions are both temporal coherence and spatial feasibility. Third, we propose two different energy-aware denoising strategies that dynamically adapt denoising steps based on action quality assessment. These strategies ensure the generation of high-quality actions while maintaining superior computational efficiency compared to fixed-step denoising approaches. Experimental results demonstrate that EnergyAction effectively transfers unimanual knowledge to bimanual tasks, achieving superior performance on both simulated and real-world tasks with minimal bimanual data.

Mingchen Song, Xiang Deng, Jie Wei +3

Harbin Institute of Technology (Shenzhen) · PengCheng Laboratory · Shenzhen Loop Area Institute

Bimanual Robotic ManipulationPolicy LearningTransfer LearningEnergy-Based Models
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Mar 6, 2026

Stochastic Attention via Langevin Dynamics on the Modern Hopfield Energy

Attention heads retrieve: given a query, they return a weighted average of stored values. We showed that this computation is one step of gradient descent on the modern Hopfield energy, and that Langevin sampling from the corresponding Boltzmann distribution yielded stochastic attention, a training-free sampler controlled by a single temperature parameter. Lowering the temperature gave exact retrieval; raising it gave open-ended generation. Because the energy gradient equals the attention map, no score network, training loop, or learned model was required, making the approach particularly suited to the low-data regime where learned generative models are starved of training signal. We derived an entropy inflection condition that identified the retrieval-to-generation transition temperature for any memory geometry and validated the sampler on five domains spanning two orders of magnitude in dimension. A single Boolean mask on the attention softmax, identical to the causal mask used in transformers but applied along the memory axis rather than the sequence axis, turned the sampler into a zero-shot class-conditional generator on Olivetti faces with no retraining and no learned classifier. On MNIST digit images, stochastic attention produced samples that were markedly more novel and more diverse than the best learned baseline while matching a Metropolis-corrected gold standard. On protein sequences from a small Pfam family, the generation regime preserved amino acid composition far more faithfully than a variational autoencoder at matched novelty, indicating that the training-free score function retained family-level fidelity that learned models lost. A denoising diffusion baseline failed across all memory sizes tested, producing samples indistinguishable from isotropic noise. The approach required no architectural changes to the underlying attention mechanism.

Abdulrahman Alswaidan, Jeffrey D. Varner

R.F. Smith School of Chemical and Biomolecular Engineering Cornell University, Ithaca, NY 14850

Hopfield NetworksEnergy-Based ModelsTransformer AttentionConditional Generative ModelingLangevin Dynamics
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Mar 3, 2026

Graph Hopfield Networks: Energy-Based Node Classification with Associative Memory

We introduce Graph Hopfield Networks, whose energy function couples associative memory retrieval with graph Laplacian smoothing for node classification. Gradient descent on this joint energy yields an iterative update interleaving Hopfield retrieval with Laplacian propagation. Memory retrieval provides regime-dependent benefits: up to 2.0~pp on sparse citation networks and up to 5 pp additional robustness under feature masking; the iterative energy-descent architecture itself is a strong inductive bias, with all variants (including the memory-disabled NoMem ablation) outperforming standard baselines on Amazon co-purchase graphs. Tuning enables graph sharpening for heterophilous benchmarks without architectural changes.

Abinav Rao, Alex Wa, Rishi Athavale

Node ClassificationHopfield NetworksEnergy-Based Models
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Mar 3, 2026

Thermodynamic Regulation of Finite-Time Gibbs Training in Energy-Based Models: A Restricted Boltzmann Machine Study

Restricted Boltzmann Machines (RBMs) are typically trained using finite-length Gibbs chains under a fixed sampling temperature. This practice implicitly assumes that the stochastic regime remains valid as the energy landscape evolves during learning. We argue that this assumption can become structurally fragile under finite-time training dynamics. This fragility arises because, in nonconvex energy-based models, fixed-temperature finite-time training can generate admissible trajectories with effective-field amplification and conductance collapse. As a result, the Gibbs sampler may asymptotically freeze, the negative phase may localize, and, without sufficiently strong regularization, parameters may exhibit deterministic linear drift. To address this instability, we introduce an endogenous thermodynamic regulation framework in which temperature evolves as a dynamical state variable coupled to measurable sampling statistics. Under standard local Lipschitz conditions and a two-time-scale separation regime, we establish global parameter boundedness under strictly positive L2 regularization. We further prove local exponential stability of the thermodynamic subsystem and show that the regulated regime mitigates inverse-temperature blow-up and freezing-induced degeneracy within a forward-invariant neighborhood. Experiments on MNIST demonstrate that the proposed self-regulated RBM substantially improves normalization stability and effective sample size relative to fixed-temperature baselines, while preserving reconstruction performance. Overall, the results reinterpret RBM training as a controlled non-equilibrium dynamical process rather than a static equilibrium approximation.

Görkem Can Süleymanoğlu

Owner and Software Developer, Kuanka Publishing LLC, Turkey. · Ph.D. Student in Economics, Selçuk University, Institute of Social Sciences, Turkey.

Restricted Boltzmann MachinesEnergy-Based ModelsStatistical Physics of LearningMarkov Chain Monte Carlo
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Feb 26, 2026

GeoWorld: Geometric World Models

Energy-based predictive world models provide a powerful approach for multi-step visual planning by reasoning over latent energy landscapes rather than generating pixels. However, existing approaches face two major challenges: (i) their latent representations are typically learned in Euclidean space, neglecting the underlying geometric and hierarchical structure among states, and (ii) they struggle with long-horizon prediction, which leads to rapid degradation across extended rollouts. To address these challenges, we introduce GeoWorld, a geometric world model that preserves geometric structure and hierarchical relations through a Hyperbolic JEPA, which maps latent representations from Euclidean space onto hyperbolic manifolds. We further introduce Geometric Reinforcement Learning for energy-based optimization, enabling stable multi-step planning in hyperbolic latent space. Extensive experiments on CrossTask and COIN demonstrate around 3% SR improvement in 3-step planning and 2% SR improvement in 4-step planning compared to the state-of-the-art V-JEPA 2. Project website: https://steve-zeyu-zhang.github.io/GeoWorld.

Zeyu Zhang, Danning Li, Ian Reid +1

ANU · MBZUAI

World Model LearningWorld ModelsWorld Model-Based PlanningEnergy-Based ModelsReinforcement Learning
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Feb 22, 2026

Kaiwu-PyTorch-Plugin: Bridging Deep Learning and Photonic Quantum Computing for Energy-Based Models and Active Sample Selection

This paper introduces the Kaiwu-PyTorch-Plugin (KPP) to bridge Deep Learning and Photonic Quantum Computing across multiple dimensions. KPP integrates the Coherent Ising Machine into the PyTorch ecosystem, addressing classical inefficiencies in Energy-Based Models. The framework facilitates quantum integration in three key aspects: accelerating Boltzmann sampling, optimizing training data via Active Sampling, and constructing hybrid architectures like QBM-VAE and Q-Diffusion. Empirical results on single-cell and OpenWebText datasets demonstrate KPPs ability to achieve SOTA performance, validating a comprehensive quantum-classical paradigm.

Hongdong Zhu, Qi Gao, Yin Ma +6

Beijing QBoson Quantum Technology Co., Ltd. · Chinese Academy of Sciences

Hybrid Quantum-Classical MLQuantum Machine LearningQuantum Generative ModelingEnergy-Based Models
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Feb 21, 2026

Spilled Energy in Large Language Models

We reinterpret the final Large Language Model (LLM) softmax classifier as an Energy-Based Model (EBM), decomposing the sequence-to-sequence probability chain into multiple interacting EBMs at inference. This principled approach allows us to track "energy spills" during decoding, which we empirically show correlate with factual errors, biases, and failures. Similar to Orgad et al. (2025), our method localizes the exact answer token and subsequently tests for hallucinations. Crucially, however, we achieve this without requiring trained probe classifiers or activation ablations. Instead, we introduce two completely training-free metrics derived directly from output logits: spilled energy, which captures the discrepancy between energy values across consecutive generation steps that should theoretically match, and marginalized energy, which is measurable at a single step. Evaluated on nine benchmarks across state-of-the-art LLMs (including LLaMA, Mistral, and Gemma) and on synthetic algebraic operations (Qwen3), our approach demonstrates robust, competitive hallucination detection and cross-task generalization. Notably, these results hold for both pretrained and instruction-tuned variants without introducing any training overhead. Code available at: github.com/OmnAI-Lab/spilled-energy

Adrian Robert Minut, Hazem Dewidar, Iacopo Masi

Sapienza University of Rome, Italy · OmnAI Lab GLADIA

LLM ReliabilityLLM Hallucination DetectionEnergy-Based Models
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Dec 4, 2025

An Energy-Based Mechanism for Compositional Behavior

Flexible intelligence relies on the ability to reuse previously acquired behaviors and combine them differently as circumstances change. In biological and artificial systems, this ability is often attributed to gating mechanisms that determine how much each available behavior should contribute at a given time. Yet these gating rules, the dynamics that compute them, and the neural circuits that may implement them are usually introduced separately, leaving unclear whether they reflect a common underlying principle. Here, we show that they can all be derived from a single variational principle for behavioral composition. The resulting mechanism naturally gives rise to softmax gating, evolves as an energy-based dynamical system with guaranteed convergence, and admits a recurrent neural network instantiation featuring context-dependent and local interactions. Across collective behavior, human decision-making, and layered control, the same mechanism reproduces characteristic behavioral patterns, provides interpretable accounts of how different behaviors are combined, and matches or outperforms established approaches. These results provide a unified account of how behavioral composition can emerge from a common principle, with implications for understanding flexible behavior in natural systems and for designing artificial agents that can adapt by recombining existing capabilities.

Francesca Rossi, Veronica Centorrino, Francesco Bullo +1

Scuola Superiore Meridionale, Italy. · ETH, Zürich. · Center for Control, Dynamical Systems, and Computation, UC Santa Barbara, CA, USA. +1

Energy-Based Models
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Dec 2, 2025

Training Energy-Based Models with Non-MCMC Samplers and Efficient Temperature Estimation

Efficient sampling from Boltzmann distributions over discrete variables is a fundamental operation in a wide range of applications. While fast non-MCMC samplers have recently emerged as promising alternatives to conventional MCMC methods, their practical use for probabilistic learning remains hindered by the difficulty of estimating the effective temperature of the generated samples. In this work, we begin by introducing Langevin simulated bifurcation (LSB), a Boltzmann sampler that enables fast and parallel sampling with accuracy comparable to sequential MCMC methods. To address the challenge of unknown effective temperature, we propose conditional expectation matching (CEM), an efficient estimation method applicable to energy-based models (EBMs) with exploitable conditional independence structures. Building on these components, we further develop a learning framework, termed sampler adaptive learning (SAL), which adaptively adjusts the model temperature to match that of the distribution induced by fast non-MCMC sampling. We demonstrate the effectiveness of LSB, CEM, and SAL on semi-restricted Boltzmann machines (SRBMs), a class of EBMs that are difficult to train using conventional approaches. LSB achieves orders-of-magnitude acceleration over Gibbs sampling while maintaining comparable or higher accuracy, and CEM enables accurate temperature estimation of the resulting distribution with negligible computational overhead. As a consequence, SAL enables efficient training of SRBMs and outperforms conventional Boltzmann machine learning methods on synthetic spin-glass datasets. In addition, the trained models achieve strong performance across multiple tasks. These results establish LSB as a fast and accurate Boltzmann sampler and provide key insights that enable practical applications of fast non-MCMC sampling methods via efficient temperature estimation with CEM.

Kentaro Kubo, Hayato Goto

Corporate Laboratory, Toshiba Corporation, Kawasaki, Kanagawa 212-8582, Japan · RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan

Boltzmann SamplingEnergy-Based Models
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Oct 14, 2025

Learning Latent Energy-Based Models via Interacting Particle Langevin Dynamics

We develop interacting particle algorithms for learning latent variable models with energy-based priors. To do so, we leverage recent developments in particle-based methods for solving maximum marginal likelihood estimation (MMLE) problems. Specifically, we provide a continuous-time framework for learning latent energy-based models, by defining stochastic differential equations (SDEs) that provably solve the MMLE problem. We obtain a practical algorithm as a discretisation of these SDEs and provide theoretical guarantees for the convergence of the proposed algorithm. Finally, we empirically validate the effectiveness of our method on synthetic and image datasets and demonstrate that using a particle based approach offers significant improvement in computational efficiency.

Joanna Marks, Tim Y. J. Wang, O. Deniz Akyildiz

Department of Mathematics, Imperial College London

Parameter EstimationLatent Variable ModelsEnergy-Based ModelsMaximum Marginal Likelihood Estimation
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Oct 2, 2025

Uniform-in-time convergence bounds for Persistent Contrastive Divergence algorithms

We propose a continuous-time formulation of a noisy persistent contrastive divergence (PCD)-like method for maximum likelihood estimation (MLE) of unnormalised densities. Our approach couples parameter updates and sampling of the parametrised density in a multiscale system of stochastic differential equations (SDEs). From this formulation, we derive non-asymptotic bounds for weak test-function errors between the resulting numerical schemes and the MLE point target. The error is decomposed into numerical discretisation, slow-fast averaging, and finite-temperature concentration terms. We also introduce an efficient implementation based on explicit stabilized integrators and establish corresponding long-time error estimates. This leads to a novel method for training energy-based models (EBMs) with quantitative error guarantees.

Paul Felix Valsecchi Oliva, O. Deniz Akyildiz, Andrew Duncan

Department of Mathematics, Imperial College London

Parameter EstimationStochastic ApproximationEnergy-Based ModelsConvergence Analysis
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Jul 16, 2025

CoCre-Sam (Kokkuri-san): Modeling Ouija Board as Collective Langevin Dynamics Sampling from Fused Language Models

Collective human activities like using an Ouija board (or Kokkuri-san) often produce emergent, coherent linguistic outputs unintended by any single participant. While psychological explanations such as the ideomotor effect exist, a computational understanding of how decentralized, implicit linguistic knowledge fuses through shared physical interaction remains elusive. We introduce CoCre-Sam (Collective-Creature Sampling), a framework modeling this phenomenon as collective Langevin dynamics sampling from implicitly fused language models. Each participant is represented as an agent associated with an energy landscape derived from an internal language model reflecting linguistic priors, and agents exert stochastic forces based on local energy gradients. We theoretically prove that the collective motion of the shared pointer (planchette) corresponds to Langevin MCMC sampling from the sum of individual energy landscapes, representing fused collective knowledge. Simulations validate that CoCre-Sam dynamics effectively fuse different models and generate meaningful character sequences, while ablation studies confirm the essential roles of collective interaction and stochasticity. Altogether, CoCre-Sam provides a novel computational mechanism linking individual implicit knowledge, embodied collective action, and emergent linguistic phenomena, grounding these complex interactions in the principles of probabilistic sampling.

Tadahiro Taniguchi, Masatoshi Nagano, Haruumi Omoto +1

Graduate School of Informatics, Kyoto University, Japan · Ritsumeikan University, Japan · University of Reading, UK

Emergent CommunicationLarge Language Model-Based Social SimulationLangevin DynamicsMarkov Chain Monte CarloEnergy-Based Models
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Jun 30, 2025

Neural Langevin Machine: a local asymmetric learning rule can be creative

Fixed points of recurrent neural networks can be leveraged to store and generate information. These fixed points are captured by the Boltzmann-Gibbs measure, which leads to neural Langevin dynamics that relax to those fixed points for generative learning of a real dataset. We call this type of generative model a neural Langevin machine, which derives an asymmetric and firing-rate-speed-adjusted learning rule requiring only local neural signals, thereby bearing biological relevance in terms of local predictive learning. An out-of-equilibrium regime of the generative process is revealed, together with a memorization-to-generalization transition with increasing training data size. The neuro-inspired machine can also realize a continuous exploration of the phase space for different kinds of generative images and can denoise a corrupted image as well.

Zhendong Yu, Weizhong Huang, Haiping Huang

PMI Lab, School of Physics, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China · Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China

Neural Network GeneralizationImage GenerationRecurrent Neural NetworksEnergy-Based ModelsGenerative ModelingLocal Learning
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May 19, 2025

Score-Based Training for Energy-Based TTS Models

Noise contrastive estimation (NCE) is a popular method for training energy-based models (EBM) with intractable normalisation terms. The key idea of NCE is to learn by comparing unnormalised log-likelihoods of the reference and noisy samples, thus avoiding explicitly computing normalisation terms. However, NCE critically relies on the quality of noisy samples. Recently, sliced score matching (SSM) has been popularised by closely related diffusion models (DM). Unlike NCE, SSM learns a gradient of log-likelihood, or score, by learning distribution of its projections on randomly chosen directions. However, both NCE and SSM disregard the form of log-likelihood function, which is problematic given that EBMs and DMs make use of first-order optimisation during inference. This paper proposes a new criterion that learns scores more suitable for first-order schemes. Experiments contrasts these approaches for training EBMs.

Wanli Sun, Anton Ragni

School of Computer Science, University of Sheffield, Sheffield, UK

Energy-Based ModelsScore MatchingScore-Based Generative ModelingTTS Synthesis
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