电网优化

Predictive Coding (PC) offers a brain-inspired alternative to backpropagation for neural network training, described as a physical system minimizing its internal energy. While ideally suited for analog implementation, such hardware does not exist yet, and thus, in practice, PC is predominantly *digitally simulated*, requiring excessive amounts of compute while struggling to scale to deeper architectures. This paper reformulates PC to overcome this hardware-algorithm mismatch. First, we uncover how the canonical state-based formulation of PC (sPC) is, by design, deeply inefficient in digital simulation, inevitably resulting in exponential signal decay that stalls the entire numerical process. Then, to overcome this fundamental limitation, we introduce error-based PC (ePC), a novel reparameterization of PC which does not suffer from signal decay. Though no longer directly implementable in analog, ePC numerically computes *exact* PC weights gradients and runs orders of magnitude faster than sPC. Experiments across multiple architectures and datasets demonstrate that ePC matches backpropagation's performance even for deeper models where sPC struggles. Besides practical improvements, our work provides theoretical insight into PC dynamics and establishes a foundation for scaling PC-based learning to deeper architectures in digital simulation and beyond.

Iterative generative policies, such as diffusion models and flow matching, offer superior expressivity for continuous control but complicate Maximum Entropy Reinforcement Learning because their action log-densities are not directly accessible. To address this, we propose \textbf{Field Least-Energy Actor-Critic (FLAC)}, a likelihood-free framework that regulates policy stochasticity by penalizing the kinetic energy of the velocity field. Our key insight is to formulate policy optimization as a Generalized Schr\"odinger Bridge (GSB) problem relative to a high-entropy reference process (e.g., uniform). Under this view, the maximum-entropy principle emerges naturally as staying close to a high-entropy reference while optimizing return, without requiring explicit action densities. In this framework, kinetic energy serves as a physically grounded proxy for divergence from the reference: minimizing path-space energy bounds the deviation of the induced terminal action distribution. Building on this view, we derive an energy-regularized policy iteration scheme and a practical off-policy algorithm that automatically tunes the kinetic energy via a Lagrangian dual mechanism. Empirically, FLAC achieves superior or comparable performance on high-dimensional benchmarks relative to strong baselines, while avoiding explicit density estimation.

The learning rate in stochastic gradient methods is a critical hyperparameter that is notoriously costly to tune via standard grid search, especially for training modern large-scale models with billions of parameters. We identify a theoretical advantage of learning rate annealing schemes that decay the learning rate to zero at a polynomial rate, such as the widely-used cosine schedule, by demonstrating their increased robustness to initial parameter misspecification due to a coarse grid search. We present an analysis in a stochastic convex optimization setup demonstrating that the convergence rate of stochastic gradient descent with annealed schedules depends *sublinearly* on the multiplicative misspecification factor $\rho$ (i.e., the grid resolution), achieving a rate of $O(\rho^{1/(2p+1)}/\sqrt{T})$ where $p$ is the degree of polynomial decay and $T$ is the number of steps. This is in contrast to the $O(\rho/\sqrt{T})$ rate obtained under the inverse-square-root and fixed stepsize schedules, which depend linearly on $\rho$. Experiments confirm the increased robustness compared to tuning with a fixed stepsize, that has significant implications for the computational overhead of hyperparameter search in practical training scenarios.

Parameter-free stochastic optimization aims to design algorithms that are agnostic to the underlying problem parameters while still achieving convergence rates competitive with optimally tuned methods. While some parameter-free methods do not require the specific values of the problem parameters, they still rely on prior knowledge, such as the lower or upper bounds of them. We refer to such methods as "partially parameter-free". In this work, we target achieving "*fully* parameter-free" methods, i.e., the algorithmic inputs do not need to satisfy any *unverifiable* condition related to the true problem parameters. We propose a powerful and general *grid search* framework, named GRASP, with a novel *self-bounding* analysis technique that effectively determines the search ranges of parameters, in contrast to previous work. Our method demonstrates generality in: (i) the non-convex case, where we propose a fully parameter-free method that achieves near-optimal convergence rate, up to logarithmic factors; (ii) the convex case, where our parameter-free methods are competitive with strong performance in terms of acceleration and universality. Finally, we contribute a sharper guarantee for the model ensemble, a final step of the grid search framework, under interpolated variance characterization.

Predictive coding networks are neural models that perform inference through an iterative energy minimization process, whose operations are local in space and time. While effective in shallow architectures, they suffer significant performance degradation beyond five to seven layers. In this work, we show that this degradation is caused by exponentially imbalanced errors between layers during weight updates, and by predictions from the previous layers not being effective in guiding updates in deeper layers. Furthermore, when training models with skip connections, the energy propagated by the residuals reaches higher layers faster than that propagated by the main pathway, affecting test accuracy. We address the first issue by introducing a novel precision-weighted optimization of latent variables that balances error distributions during the relaxation phase, the second issue by proposing a novel weight update mechanism that reduces error accumulation in deeper layers, and the third one by using auxiliary neurons that slow down the propagation of the energy in the residual connections. Empirically, our methods achieve performance comparable to backpropagation on deep models such as ResNets, opening new possibilities for predictive coding in complex tasks.

Large language model inference is often bounded by memory footprint and bandwidth in resource-constrained deployments, making quantization fundamental to efficient serving. While post-training quantization (PTQ) maintains high fidelity at 4-bit, it deteriorates at 2-3 bits. In essence, existing methods enforce a shape-invariant quantization grid (e.g., the fixed uniform intervals of UINT2) for each group, severely restricting the feasible set for error minimization. To address this, we propose Bit-Plane Decomposition Quantization (BPDQ), which constructs a variable quantization grid via bit-planes and scalar coefficients, and iteratively refines them using second-order information while progressively compensating for quantization errors to minimize output discrepancy. In the 2-bit regime, BPDQ enables serving Qwen2.5-72B on a single RTX 3090 with 83.85\% GSM8K accuracy (vs. 90.83\% at 16-bit). Moreover, we theoretically show that the variable grid expands the feasible set, and that the quantization process consistently aligns with the optimization objective in Hessian-induced geometry. The code is available at github.com/KingdalfGoodman/BPDQ.

Energy-based models (EBMs) provide a flexible framework for generative models with strong distribution modeling capabilities. Nevertheless, their broader adoption has been limited by the difficulty of stable and efficient training. In this paper, we propose a unified and efficient latent-guided cooperative EBM that leverages informative target latent variables to guide the joint energy in capturing both data distribution and semantic structure, along with a cooperative generator designed for effective MCMC initialization. Our joint space optimization only requires MCMC sampling in the data space, and allows the energy to learn semantic data–latent relationships directly from real data. Experiments show our method improves generation quality and training stability with fewer resources, and performs effectively across multiple downstream tasks.

3D Gaussian Splatting (3DGS) has been widely adopted for scene reconstruction, where training inherently constitutes a highly coupled and non-convex optimization problem. Recent works commonly incorporate geometric priors, such as LiDAR measurements, either for initialization or as training constraints, with the goal of improving photometric reconstruction quality. However, in large-scale outdoor scenarios, such geometric supervision is often spatially incomplete and uneven, which limits its effectiveness as a reliable prior and can even be detrimental to the final reconstruction. To address this challenge, we model partially observable geometry as a continuous energy field induced by geometric evidence and propose EnerGS. Rather than enforcing geometry as a hard constraint, EnerGS provides a soft geometric guidance for the optimization of Gaussian primitives, allowing geometric information to steer the optimization process without directly restricting the solution space. Extensive experiments on large-scale outdoor scenes demonstrate that, under both sparse multi-view and monocular settings, EnerGS consistently improves photometric quality and geometric stability, while effectively mitigating overfitting during 3DGS training. The codebase is publicly available at: https://github.com/ucla-mobility/EnerGS.

Flow-based generative models can be viewed through a physics lens: sampling transports a particle from noise to data by integrating a learned velocity field, and each sample corresponds to a trajectory with its own dynamical effort. Motivated by classical mechanics, we introduce Kinetic Path Energy (KPE), an action-like, per-sample diagnostic that measures the accumulated kinetic effort along an ordinary differential equation (ODE) trajectory. Empirically, KPE exhibits two robust correspondences: {i} higher KPE predicts stronger semantic fidelity; {ii} high-KPE trajectories land in sparse representation regions. We further provide theoretical guarantees linking trajectory energy to data sparsity. Paradoxically, this correlation is non-monotonic. At sufficiently high energy, generation can degenerate into memorization. Leveraging the closed-form formula of empirical flow matching, we show that extreme energies drive trajectories toward near-copies of training examples. This yields a Goldilocks principle and motivates Kinetic Trajectory Shaping (KTS), a training-free two-phase inference strategy that boosts early motion and enforces a late-time soft landing, reducing memorization and improving generation quality across benchmark tasks.

Continual learning (CL) aims to acquire new knowledge from a non-stationary data stream while retaining performance on previously learned tasks. Memory-based replay methods mitigate catastrophic forgetting by storing and revisiting past samples, but their effectiveness is fundamentally constrained by limited memory capacity, as each stored example represents only a single data instance. In this work, we propose data reassembly for CL, a new paradigm that significantly increases memory efficiency by reassembling composite replay samples from existing training data. Instead of storing raw training examples, we partition the current task training data into elementary patches and dynamically reassemble them into coherent replay instances through an energy-based optimization framework. The proposed objective jointly enforces semantic compatibility with target labels and global consistency among assembled patches. To make this optimization tractable, we derive an efficient variational inference algorithm that constructs a compact yet diverse set of reassembled samples for replay. Extensive theoretical analysis and experiments across multiple CL benchmarks demonstrate that data reassembly consistently outperforms existing memory-based approaches, achieving stronger retention of past knowledge while maintaining competitive computational efficiency.

Protein mutation effect prediction is fundamental to protein engineering and disease variant interpretation, yet experimentally measured mutation data remain accurate but extremely sparse. To provide scalable supplementary mutation signals, we construct a PDB-wide mutation augmentation dataset that exhaustively enumerates single-site substitutions on experimentally resolved protein structures and aligns mutation signals from physics-based energy models, protein language models, and inverse folding models. Large-scale analysis under a unified mutation preference representation reveals substantial differences in the consistency, concentration, and substitution patterns of mutation distributions across models, indicating that disagreement is pervasive and reflects conflicting inductive biases rather than random noise. Motivated by these observations, we propose an unsupervised multi-source mutation preference distillation framework that learns from relative mutation preferences while explicitly modeling cross-source disagreement. Without using any experimental mutation labels during training, our approach achieves the best overall performance among the evaluated zero-shot baselines and naive multi-source fusion strategies on ProteinGym. We release the dataset and evaluation pipeline to support reproducible studies of protein mutation effects.

Generative diffusion models can provide powerful prior probability models for inverse problems in imaging, but existing implementations suffer from two key limitations: $(i)$ the prior density is represented implicitly, and $(ii)$ they rely on likelihood approximations that introduce sampling biases. We address these challenges by introducing a new energy-based model trained for denoising with a covariance-based regularization term that enforces consistency across different measurement conditions. The trained model can compute normalized posterior densities for diverse linear inverse problems, without additional retraining or fine tuning. In addition to preserving the sampling capabilities of diffusion models, this enables previously unavailable capabilities: energy-guided adaptive sampling that adjusts schedules on-the-fly, unbiased Metropolis-Hastings correction steps, and blind estimation of the degradation operator via Bayes rule. We validate the method on multiple datasets (ImageNet, CelebA, AFHQ) and tasks (inpainting, deblurring), demonstrating competitive or superior performance to established baselines.

We propose an OT-guided sampling framework for solving graph combinatorial optimization through exact multilinear relaxation. Graph combinatorial optimization problems can be written as quadratic unconstrained binary optimization(QUBO). Leveraging a classical result in combinatorial optimization, we obtain a continuous multi-linear relaxation of QUBO that is exact, in the sense that it preserves the optimal binary solutions. The challenge is that the resulting energy landscape is highly nonconvex. We address this by treating the objective as an energy function and optimizing via sampling from the induced Boltzmann distribution to escape poor local optima. Viewing sampling as transporting a simple reference distribution to the target distribution, we use optimal transport to characterize more efficient probability flow and derive a stochastic optimal control problem whose solution yields an optimal sampling dynamics. We parameterize the control policy with graph neural networks to approximate the optimal control. Experiments show improved solution quality and efficiency over strong combinatorial and learning-based baselines.

A key challenge in deriving unified neural solvers for combinatorial optimization (CO) is efficient generalization of models between one set of tasks to new tasks not used during the initial training process. To address it, we first establish a new GNN encoder, which uses a GCON module as a form of expressive message passing together with energy-based unsupervised loss functions. This model achieves high performance (often comparable with state-of-the-art results) across multiple CO tasks when trained individually on each task. We then leverage knowledge from the computational reducibility literature to propose pretraining and fine-tuning strategies to transfer effectively (a) between MVC, MIS and MaxClique, and (b) in a multi-task learning setting that additionally incorporates MaxCut, MDS and graph coloring. Additionally, in a leave-one-out, multi-task learning setting, we observe that pretraining on all but one task almost always leads to faster convergence on the remaining task when fine-tuning while avoiding negative transfer. Our findings indicate that learning common representations across multiple graph CO problems is viable through the use of expressive message passing coupled with pretraining strategies that are informed by the polynomial reduction literature, thereby taking an important step towards enabling the development of foundational models for neural CO.

A fundamental challenge in data science is to match disparate point sets with each other. While optimal transport efficiently minimizes point *displacements* under a bijectivity constraint, it is inherently sensitive to rotations. Conversely, minimizing *distortions* via the Gromov-Wasserstein (GW) framework addresses this limitation but introduces a non-convex, computationally demanding optimization problem. In this work, we identify a broad class of distortion penalties that reduce to a simple alignment problem within a lifted feature space. Leveraging this insight, we introduce an iterative GW solver with a linear memory footprint and quadratic (rather than cubic) time complexity. Our method is differentiable, comes with strong theoretical guarantees, and scales to hundreds of thousands of points in minutes. This efficiency unlocks a wide range of geometric applications and enables the exploration of the GW energy landscape, whose local minima encode the symmetries of the matching problem.

Deep learning approaches typically require prohibitive amounts of data to approximate known-constraint Exact Cover tasks, while existing neuro-symbolic methods often face training infeasibility and scalability bottlenecks. To bridge this divide, we propose the Hypergraph Optimization Network (HONet), an end-to-end framework integrating a structure-preserving Deep Residual Hypergraph Encoder with a differentiable fixed-constraint Quadratic Programming layer. By adopting a ``Fixed Polytope'' paradigm guided by the Geometric Consistency Loss, HONet explicitly shapes the objective landscape, encouraging the valid discrete solution to align with the global energy minimum. Empirical results show that HONet rapidly achieves 100\% accuracy on $9 \times 9$ Sudoku using limited samples, exhibiting superior data efficiency over baselines while maintaining exceptional robustness in highly sparse regimes and additional tasks.

Optimization modeling underpins decision-making in logistics, manufacturing, energy, and finance, yet translating natural-language requirements into correct optimization formulations and solver-executable code remains labor-intensive. Although large language models (LLMs) have been explored for this task, evaluation is still dominated by toy-sized or synthetic benchmarks, masking the difficulty of industrial problems with $10^{3}$--$10^{6}$ (or more) variables and constraints. A key bottleneck is the lack of benchmarks that align natural-language specifications with reference formulations/solver code grounded in real optimization models. To fill in this gap, we introduce MIPLIB-NL, built via a structure-aware reverse construction methodology from real mixed-integer linear programs in MIPLIB~2017. Our pipeline (i) recovers compact, reusable model structure from flat solver formulations, (ii) reverse-generates natural-language specifications explicitly tied to this recovered structure under a unified model--data separation format, and (iii) performs iterative semantic validation through expert review and human--LLM interaction with independent reconstruction checks. This yields 223 one-to-one reconstructions that preserve the mathematical content of the original instances while enabling realistic natural-language-to-optimization evaluation. Experiments show substantial performance degradation on MIPLIB-NL for systems that perform strongly on existing benchmarks, exposing failure modes invisible at toy scale.