Energy Grid Optimization

We study the Quality of Service Degradation (QoSD) problem, in which an adversary perturbs edge weights to degrade network performance. This setting arises in both network infrastructures and distributed ML systems, where communication quality, not just connectivity, determines functionality. While classical methods rely on combinatorial optimization, and recent ML approaches address only restricted linear variants with small-size networks, no prior model directly tackles the QoSD problem under nonlinear edge-weight functions. This work proposes Hephaestus, a self-reinforcing generative framework that synthesizes feasible solutions in latent space, to fill this gap. Our method includes three phases: (1) Forge: a Predictive Path-Stressing (PPS) algorithm that uses graph learning and approximation to produce feasible solutions with performance guarantee, (2) Morph: a new theoretically grounded training paradigm for Mixture of Conditional VAEs guided by an energy-based model to capture solution feature distributions, and (3) Refine: a reinforcement learning agent that explores this space to generate progressively near-optimal solutions using our designed differentiable reward function. Experiments on both synthetic and real-world networks show that our approach consistently outperforms classical and ML baselines, particularly in scenarios with nonlinear cost functions where traditional methods fail to generalize.

**Diffusion models** achieve remarkable generative performance but are hampered by slow, iterative inference. Model distillation seeks to train a fast student generator. **Variational Score Distillation (VSD)** offers a principled KL-divergence minimization framework for this task. This method cleverly avoids computing the teacher model's Jacobian, but its student gradient relies on the score of the student's own noisy marginal distribution, $\nabla\_{\mathbf{x}\_t} \log p\_{\phi,t}(\mathbf{x}\_t)$. VSD thus requires approximations, such as training an auxiliary network to estimate this score. These approximations can introduce biases, cause training instability, or lead to an incomplete match of the target distribution, potentially focusing on conditional means rather than broader distributional features. We introduce **VarFlow**, a method based on a **Score-Rule Variational Distillation (SRVD)** framework. VarFlow trains a one-step generator $g_{\phi}(\mathbf{z})$ by directly minimizing an energy distance (derived from the strictly proper energy score) between the student's induced noisy data distribution $p_{\phi,t}(\mathbf{x}_t)$ and the teacher's target noisy distribution $q_t(\mathbf{x}_t)$. This objective is estimated entirely using samples from these two distributions. Crucially, VarFlow bypasses the need to compute or approximate the intractable student score. By directly matching the full noisy marginal distributions, VarFlow aims for a more comprehensive and robust alignment between student and teacher, offering an efficient and theoretically grounded path to high-fidelity one-step generation.

Conditional decision generation with diffusion models has shown powerful competitiveness in reinforcement learning (RL). Recent studies reveal the relation between energy-function-guidance diffusion models and constrained RL problems. The main challenge lies in estimating the intermediate energy, which is intractable due to the log-expectation formulation during the generation process. To address this issue, we propose the Analytic Energy-guided Policy Optimization (AEPO). Specifically, we first provide a theoretical analysis and the closed-form solution of the intermediate guidance when the diffusion model obeys the conditional Gaussian transformation. Then, we analyze the posterior Gaussian distribution in the log-expectation formulation and obtain the target estimation of the log-expectation under mild assumptions. Finally, we train an intermediate energy neural network to approach the target estimation of log-expectation formulation. We apply our method in 30+ offline RL tasks to demonstrate the effectiveness of our method. Extensive experiments illustrate that our method surpasses numerous representative baselines in D4RL offline reinforcement learning benchmarks.

We introduce Linearly Constrained Diffusion Implicit Models (CDIM), a fast and accurate approach to solving noisy linear inverse problems using diffusion models. Traditional diffusion-based inverse methods rely on numerous projection steps to enforce measurement consistency in addition to unconditional denoising steps. CDIM achieves a 10–50× reduction in projection steps by dynamically adjusting the number and size of projection steps to align a residual measurement energy with its theoretical distribution under the forward diffusion process. This adaptive alignment preserves measurement consistency while substantially accelerating constrained inference. For noise-free linear inverse problems, CDIM exactly satisfies the measurement constraints with few projection steps, even when existing methods fail. We demonstrate CDIM’s effectiveness across a range of applications, including super-resolution, denoising, inpainting, deblurring, and 3D point cloud reprojection.

Searching through chemical space is an exceptionally challenging problem because the number of possible molecules grows combinatorially with the number of atoms. Large, autoregressive models trained on databases of chemical compounds have yielded powerful generators, but we still lack robust strategies for generating molecules with desired properties. This molecular search problem closely resembles the "alignment" problem for large language models, though for many chemical tasks we have a specific and easily evaluable reward function. Here, we introduce an algorithm called energy rank alignment (ERA) that leverages an explicit reward function to produce a gradient-based objective that we use to optimize autoregressive policies. We show theoretically that this algorithm is closely related to proximal policy optimization (PPO) and direct preference optimization (DPO), but has a minimizer that converges to an ideal Gibbs-Boltzmann distribution with the reward playing the role of an energy function. Furthermore, this algorithm is highly scalable, does not require reinforcement learning, and performs well relative to DPO when the number of preference observations per pairing is small. We deploy this approach to align molecular transformers and protein language models to generate molecules and protein sequences, respectively, with externally specified properties and find that it does so robustly, searching through diverse parts of chemical space.

While energy-based models have recently proven to be a powerful framework for learning to reason with neural networks, their practical application is still limited by computational cost. That is, existing methods for energy-based iterative reasoning suffer from computational bottlenecks by relying on expensive optimization routines during training and especially during inference. Furthermore, these routines may not always converge to minimal energy states, potentially leading to suboptimal reasoning. To address these limitations, we propose a novel and efficient algorithm for energy-based iterative reasoning based on a difference-of-convex (DC) functions approach. Our algorithm achieves a significant speedup compared to prior methods while offering theoretical convergence guarantees ensuring locally minimal energy states. In addition, we achieve state-of-the-art or superior performance on continuous reasoning tasks, as demonstrated by our experiments on multiple benchmark datasets from continuous algorithmic reasoning. As such, our method offers a leap in computational efficiency, enabling faster inference with theoretical guarantees, and hence unlocking the potential of energy-based models for iterative reasoning applications.

Existing Bayesian Optimization (BO) methods typically balance exploration and exploitation to optimize costly objective functions. However, these methods often suffer from a significant one-step bias, which may lead to convergence toward local optima and poor performance in complex or high-dimensional tasks. Recently, Black-Box Optimization (BBO) has achieved success across various scientific and engineering domains, particularly when function evaluations are costly and gradients are unavailable. Motivated by this, we propose the Reinforced Energy-Based Model for Bayesian Optimization (REBMBO), which integrates Gaussian Processes (GP) for local guidance with an Energy-Based Model (EBM) to capture global structural information. Notably, we define each Bayesian Optimization iteration as a Markov Decision Process (MDP) and use Proximal Policy Optimization (PPO) for adaptive multi-step lookahead, dynamically adjusting the depth and direction of exploration to effectively overcome the limitations of traditional BO methods. We conduct extensive experiments on synthetic and real-world benchmarks, confirming the superior performance of REBMBO. Additional analyses across various GP configurations further highlight its adaptability and robustness.

Stellarators are magnetic confinement devices under active development to deliver steady-state carbon-free fusion energy. Their design involves a high-dimensional, constrained optimization problem that requires expensive physics simulations and significant domain expertise. Recent advances in plasma physics and open-source tools have made stellarator optimization more accessible. However, broader community progress is currently bottlenecked by the lack of standardized optimization problems with strong baselines and datasets that enable data-driven approaches, particularly for quasi-isodynamic (QI) stellarator configurations, considered as a promising path to commercial fusion due to their inherent resilience to current-driven disruptions. Here, we release an open dataset of diverse QI-like stellarator plasma boundary shapes, paired with their ideal magnetohydrodynamic (MHD) equilibria and performance metrics. We generated this dataset by sampling a variety of QI fields and optimizing corresponding stellarator plasma boundaries. We introduce three optimization benchmarks of increasing complexity: (1) a single-objective geometric optimization problem, (2) a "simple-to-build" QI stellarator, and (3) a multi-objective ideal-MHD stable QI stellarator that investigates trade-offs between compactness and coil simplicity. For every benchmark, we provide reference code, evaluation scripts, and strong baselines based on classical optimization techniques. Finally, we show how learned models trained on our dataset can efficiently generate novel, feasible configurations without querying expensive physics oracles. By openly releasing the dataset (https://huggingface.co/datasets/proxima-fusion/constellaration) along with benchmark problems and baselines (https://github.com/proximafusion/constellaration), we aim to lower the entry barrier for optimization and machine learning researchers to engage in stellarator design and to accelerate cross-disciplinary progress toward bringing fusion energy to the grid.

Machine learning solutions are rapidly adopted to enable a variety of key use cases, from conversational AI assistants to scientific discovery. As the adoption of machine learning models becomes increasingly prevalent, the associated lifecycle carbon footprint is expected to increase, including both *operational carbon* from training and inference and *embodied carbon* from AI hardware manufacturing. We introduce CATransformers, the first carbon-aware co-optimization framework for Transformer-based models and hardware accelerators. By integrating both operational and embodied carbon into early-stage design space exploration, CATransformers enables sustainability-driven model architecture and hardware accelerator co-design that reveals fundamentally different trade-offs than latency- or energy-centric approaches. Evaluated across a range of Transformer models, CATransformers consistently demonstrates the potential to reduce total carbon emissions --by up to 30\% -- while maintaining accuracy and latency. We further highlight its extensibility through a focused case study on multi-modal models. Our results emphasize the need for holistic optimization methods that prioritize carbon efficiency without compromising model capability and execution time performance. The source code of CATransformers is available at https://github.com/facebookresearch/CATransformers.

Predicting changes in binding free energy ($\Delta\Delta G$) is essential for understanding protein-protein interactions, which are critical in drug design and protein engineering. However, existing methods often rely on pre-trained knowledge and heuristic features, limiting their ability to accurately model complex mutation effects, particularly higher-order and many-body interactions. To address these challenges, we propose H3-DDG, a Hypergraph-driven Hierarchical network to capture Higher-order many-body interactions across multiple scales. By introducing a hierarchical communication mechanism, H3-DDG effectively models both local and global mutational effects. Experimental results demonstrate state-of-the-art performance on multiple benchmarks. On the SKEMPI v2 dataset, H3-DDG achieves a Pearson correlation of 0.75, improving multi-point mutations prediction by 12.10%. On the challenging BindingGYM dataset, it outperforms Prompt-DDG and BA-DDG by 62.61% and 34.26%, respectively. Ablation and efficiency analyses demonstrate its robustness and scalability, while a case study on SARS-CoV-2 antibodies highlights its practical value in improving binding affinity for therapeutic design.

As the adoption of Generative AI in real-world services grow explosively, energy has emerged as a critical bottleneck resource. However, energy remains a metric that is often overlooked, under-explored, or poorly understood in the context of building ML systems. We present the ML.ENERGY Benchmark, a benchmark suite and tool for measuring inference energy consumption under realistic service environments, and the corresponding ML.ENERGY Leaderboard, which have served as a valuable resource for those hoping to understand and optimize the energy consumption of their generative AI services. In this paper, we explain four key design principles for benchmarking ML energy we have acquired over time, and then describe how they are implemented in the ML.ENERGY Benchmark. We then highlight results from the early 2025 iteration of the benchmark, including energy measurements of 40 widely used model architectures across 6 different tasks, case studies of how ML design choices impact energy consumption, and how automated optimization recommendations can lead to significant (sometimes more than 40%) energy savings without changing what is being computed by the model. The ML.ENERGY Benchmark is open-source and can be easily extended to various customized models and application scenarios.

What is the shortest path between two data points lying in a high-dimensional space? While the answer is trivial in Euclidean geometry, it becomes significantly more complex when the data lies on a curved manifold—requiring a Riemannian metric to describe the space's local curvature. Estimating such a metric, however, remains a major challenge in high dimensions. In this work, we propose a method for deriving Riemannian metrics directly from pretrained Energy-Based Models (EBMs)—a class of generative models that assign low energy to high-density regions. These metrics define spatially varying distances, enabling the computation of geodesics—shortest paths that follow the data manifold’s intrinsic geometry. We introduce two novel metrics derived from EBMs and show that they produce geodesics that remain closer to the data manifold and exhibit lower curvature distortion, as measured by alignment with ground-truth trajectories. We evaluate our approach on increasingly complex datasets: synthetic datasets with known data density, rotated character images with interpretable geometry, and high-resolution natural images embedded in a pretrained VAE latent space. Our results show that EBM-derived metrics consistently outperform established baselines, especially in high-dimensional settings. Our work is the first to derive Riemannian metrics from EBMs, enabling data-aware geodesics and unlocking scalable, geometry-driven learning for generative modeling and simulation.

Test-time adaptation (TTA) aims to preserve model performance under distribution shifts. Yet, most existing methods rely on entropy minimization for confident predictions. This paper re-examines the sufficiency of entropy minimization by analyzing its dual relationship with energy. We view energy as a proxy for likelihood, where lower energy indicates higher observability under the learned distribution. We uncover that entropy and energy are tightly associated, controlled by the model’s confidence or ambiguity, and show that simultaneous reduction of both is essential. Importantly, we reveal that entropy minimization alone neither ensures energy reduction nor supports reliable likelihood estimation, and it requires explicit discriminative guidance to reach zero entropy. To combat these problems, we propose a twofold solution. First, we introduce a likelihood-based objective grounded in energy-based models, which reshape the energy landscape to favor test samples. For stable and scalable training, we adopt sliced score matching—a sampling-free, Hessian-insensitive approximation of Fisher divergence. Second, we enhance entropy minimization with a cross-entropy that treats the predicted class as a target to promote discriminability. By counterbalancing entropy and energy through the solution of multi-objective optimization, our unified TTA, ReTTA, outperforms existing entropy- or energy-based approaches across diverse distribution shifts.

The increasing energy demands and carbon footprint of large-scale AI require intelligent workload management in globally distributed data centers. Yet progress is limited by the absence of benchmarks that realistically capture the interplay of time-varying environmental factors (grid carbon intensity, electricity prices, weather), detailed data center physics (CPUs, GPUs, memory, HVAC energy), and geo-distributed network dynamics (latency and transmission costs). To bridge this gap, we present DCcluster-Opt: an open-source, high-fidelity simulation benchmark for sustainable, geo-temporal task scheduling. DCcluster-Opt combines curated real-world datasets, including AI workload traces, grid carbon intensity, electricity markets, weather across 20 global regions, cloud transmission costs, and empirical network delay parameters with physics-informed models of data center operations, enabling rigorous and reproducible research in sustainable computing. It presents a challenging scheduling problem where a top-level coordinating agent must dynamically reassign or defer tasks that arrive with resource and service-level agreement requirements across a configurable cluster of data centers to optimize multiple objectives. The environment also models advanced components such as heat recovery. A modular reward system enables an explicit study of trade-offs among carbon emissions, energy costs, service level agreements, and water use. It provides a Gymnasium API with baseline controllers, including reinforcement learning and rule-based strategies, to support reproducible ML research and a fair comparison of diverse algorithms. By offering a realistic, configurable, and accessible testbed, DCcluster-Opt accelerates the development and validation of next-generation sustainable computing solutions for geo-distributed data centers.

We present a systematic theoretical framework that interprets masked diffusion models (MDMs) as solutions to energy minimization problems in discrete optimal transport. Specifically, we prove that three distinct energy formulations—kinetic, conditional kinetic, and geodesic energy—are mathematically equivalent under the structure of MDMs, and that MDMs minimize all three when the mask schedule satisfies a closed-form optimality condition. This unification not only clarifies the theoretical foundations of MDMs, but also motivates practical improvements in sampling. By parameterizing interpolation schedules via Beta distributions, we reduce the schedule design space to a tractable 2D search, enabling efficient post-training tuning without model modification. Experiments on synthetic and real-world benchmarks demonstrate that our energy-inspired schedules outperform hand-crafted baselines, particularly in low-step sampling settings.

We present a three-stage framework for training deep learning models specializing in antibody sequence-structure co-design. We first pre-train a language model using millions of antibody sequence data. Then, we employ the learned representations to guide the training of a diffusion model for joint optimization over both sequence and structure of antibodies. During the final alignment stage, we optimize the model to favor antibodies with low repulsion and high attraction to the antigen binding site, enhancing the rationality and functionality of the designs. To mitigate conflicting energy preferences, we extend AbDPO (Antibody Direct Preference Optimization) to guide the model toward Pareto optimality under multiple energy-based alignment objectives. Furthermore, we adopt an iterative learning paradigm with temperature scaling, enabling the model to benefit from diverse online datasets without requiring additional data. In practice, our proposed methods achieve high stability and efficiency in producing a better Pareto front of antibody designs compared to top samples generated by baselines and previous alignment techniques. Through extensive experiments, we showcase the superior performance of our methods in generating nature-like antibodies with high binding affinity.

Generalization is a key challenge in machine learning, specifically in reasoning tasks, where models are expected to solve problems more complex than those encountered during training. Existing approaches typically train reasoning models in an end-to-end fashion, directly mapping input instances to solutions. While this allows models to learn useful heuristics from data, it often results in limited generalization beyond the training distribution. In this work, we propose a novel approach to reasoning generalization by learning energy landscapes over the solution spaces of smaller, more tractable subproblems. At test time, we construct a global energy landscape for a given problem by combining the energy functions of multiple subproblems. This compositional approach enables the incorporation of additional constraints during inference, allowing the construction of energy landscapes for problems of increasing difficulty. To improve the sample quality from this newly constructed energy landscape, we introduce Parallel Energy Minimization (PEM). We evaluate our approach on a wide set of reasoning problems. Our method outperforms existing state-of-the-art methods, demonstrating its ability to generalize to larger and more complex problems. Project website can be found at: https://alexoarga.github.io/compositional_reasoning/

Inference in both brains and machines can be formalized by optimizing a shared objective: maximizing the evidence lower bound (ELBO) in machine learning, or minimizing variational free energy ($\mathcal{F}$) in neuroscience (ELBO = $-\mathcal{F}$). While this equivalence suggests a unifying framework, it leaves open how inference is implemented in neural systems. Here, we introduce FOND (*Free energy Online Natural-gradient Dynamics*), a framework that derives neural inference dynamics from three principles: (1) natural gradients on $\mathcal{F}$, (2) online belief updating, and (3) iterative refinement. We apply FOND to derive iP-VAE (*iterative Poisson variational autoencoder*), a recurrent spiking neural network that performs variational inference through membrane potential dynamics, replacing amortized encoders with iterative inference updates. Theoretically, iP-VAE yields several desirable features such as emergent normalization via lateral competition, and hardware-efficient integer spike count representations. Empirically, iP-VAE outperforms both standard VAEs and Gaussian-based predictive coding models in sparsity, reconstruction, and biological plausibility, and scales to complex color image datasets such as CelebA. iP-VAE also exhibits strong generalization to out-of-distribution inputs, exceeding hybrid iterative-amortized VAEs. These results demonstrate how deriving inference algorithms from first principles can yield concrete architectures that are simultaneously biologically plausible and empirically effective.

Lorentz-equivariant neural networks are becoming the leading architectures for high-energy physics. Current implementations rely on specialized layers, limiting architectural choices. We introduce Lorentz Local Canonicalization (LLoCa), a general framework that renders any backbone network exactly Lorentz-equivariant. Using equivariantly predicted local reference frames, we construct LLoCa-transformers and graph networks. We adapt a recent approach for geometric message passing to the non-compact Lorentz group, allowing propagation of space-time tensorial features. Data augmentation emerges from LLoCa as a special choice of reference frame. Our models achieve competitive and state-of-the-art accuracy on relevant particle physics tasks, while being $4\times$ faster and using $10\times$ fewer FLOPs.