Diffusion models (DMs) excel in image generation but suffer from slow inference and training-inference discrepancies. Although gradient-based solvers for DMs accelerate denoising inference, they often lack theoretical foundations in information transmission efficiency. In this work, we introduce an information-theoretic perspective on the inference processes of DMs, revealing that successful denoising fundamentally reduces conditional entropy in reverse transitions. This principle leads to our key insights into the inference processes: (1) data prediction parameterization outperforms its noise counterpart, and (2) optimizing conditional variance offers a reference-free way to minimize both transition and reconstruction errors. Based on these insights, we propose an entropy-aware variance optimized method for the generative process of DMs, called EVODiff, which systematically reduces uncertainty by optimizing conditional entropy during denoising. Extensive experiments on DMs validate our insights and demonstrate that our method significantly and consistently outperforms state-of-the-art (SOTA) gradient-based solvers. For example, compared to the DPM-Solver++, EVODiff reduces the reconstruction error by up to 45.5% (FID improves from 5.10 to 2.78) at 10 function evaluations (NFE) on CIFAR-10, cuts the NFE cost by 25% (from 20 to 15 NFE) for highquality samples on ImageNet-256, and improves text-to-image generation while reducing artifacts.

We present HyPINO, a multi-physics neural operator designed for zero-shot generalization across a broad class of PDEs without requiring task-specific fine-tuning. Our approach combines a Swin Transformer-based hypernetwork with mixed supervision: (i) labeled data from analytical solutions generated via the Method of Manufactured Solutions (MMS), and (ii) unlabeled samples optimized using physics-informed objectives. The model maps PDE parameterizations to target Physics-Informed Neural Networks (PINNs) and can handle linear elliptic, hyperbolic, and parabolic equations in two dimensions with varying source terms, geometries, and mixed Dirichlet/Neumann boundary conditions, including interior boundaries. HyPINO achieves strong zero-shot accuracy on seven benchmark problems from PINN literature, outperforming U-Nets, Poseidon, and Physics-Informed Neural Operators (PINO). Further, we introduce an iterative refinement procedure that treats the residual of the generated PINN as "delta PDE" and performs another forward pass to generate a corrective PINN. Summing their contributions and repeating this process forms an ensemble whose combined solution progressively reduces the error on six benchmarks and achieves a >100 lower L2 loss in the best case, while retaining forward-only inference. Additionally, we evaluate the fine-tuning behavior of PINNs initialized by HyPINO and show that they converge faster and to lower final error than both randomly initialized and Reptilemeta-learned PINNs on five benchmarks, performing on par on the remaining two. Our results highlight the potential of this scalable approach as a foundation for extending neural operators toward solving increasingly complex, nonlinear, and high-dimensional PDE problems. The code and model weights are publicly available at https://github.com/rbischof/hypino.

How precisely does circuit wiring specify function? This fundamental question is particularly relevant for modern neuroscience, as large-scale electron microscopy now enables the reconstruction of neural circuits at single-synapse resolution across many organisms. To interpret circuit function from such datasets, we must understand the extent to which the measured structure constrains dynamics. We investigate this question in the Drosophila head direction (HD) circuit, which maintains an internal heading estimate through attractor dynamics that integrate self-motion velocity cues. This circuit serves as a sensitive assay for functional specification: continuous attractor networks are theoretically known to require finely tuned wiring symmetries, whereas connectomes omit key cellular parameters such as synaptic gains, neuronal thresholds, and time constants, and reveal that biological wiring can be heterogeneous. We introduce a method that combines selfsupervised and unsupervised learning objectives to estimate unknown parameters at the level of cell types, rather than individual neurons and synapses. Starting from the raw connectivity matrix, our approach recovers a network that exhibits continuous attractor dynamics and accurately integrates a range of velocity inputs, despite minimal parameter tuning on a connectome that notably departs from the symmetric regularity of an idealized ring attractor. We characterize how deviations from the original connectome shape the space of viable solutions. We also perform in-silico ablation experiments to probe the distinct functional roles of specific cell types in the circuit, demonstrating how connectome-derived structure, when augmented with minimal, biologically grounded tuning, can replicate known physiology and elucidate circuit function.

Masked diffusion models (MDM) are powerful generative models for discrete data that generate samples by progressively unmasking tokens in a sequence. Each token can take one of two states: masked or unmasked. We observe that token sequences often remain unchanged between consecutive sampling steps; consequently, the model repeatedly processes identical inputs, leading to redundant computation. To address this inefficiency, we propose the Partial masking scheme (Prime), which augments MDM by allowing tokens to take intermediate states interpolated between the masked and unmasked states. This design enables the model to make predictions based on partially observed token information, and facilitates a fine-grained denoising process. We derive a variational training objective and introduce a simple architectural design to accommodate intermediate-state inputs. Our method demonstrates superior performance across a diverse set of generative modeling tasks. On text data, it achieves a perplexity of 15.36 on OpenWebText, outperforming previous MDM (21.52), autoregressive models (17.54), and their hybrid variants (17.58), without relying on an autoregressive formulation.

Synaptic plasticity is widely considered to be crucial to the brain's ability to learn throughout life. Decades of theoretical work have therefore been invested in deriving and designing biologically plausible learning rules capable of granting various memory abilities to neural networks. Most of these theoretical approaches optimize directly for a desired memory function; but this procedure can lead to complex, finely-tuned rules, rendering them brittle to perturbations and difficult to implement in practice. Instead, we build on recent work that automatically discovers large numbers of candidate plasticity rules operating in recurrent spiking neural networks. Surprisingly, despite the fact that these rules are selected solely to achieve network stabilization, we observe across a range of network models-- feedforward, recurrent; rate and spiking--that almost all these rules endow the network with simple forms of memory such as familiarity detection - seemingly by accident.

Estimating mutual information (MI) is a fundamental task in data science and machine learning. Existing estimators mainly rely on either highly flexible models (e.g., neural networks), which require large amounts of data, or overly simplified models (e.g., Gaussian copula), which fail to capture complex distributions. Drawing upon recent vector copula theory, we propose a principled interpolation between these two extremes to achieve a better trade-off between complexity and capacity. Experiments on state-of-the-art synthetic benchmarks and real-world data with diverse modalities demonstrate the advantages of the proposed estimator.

We show that low-rank adaptation of large-scale models suffers from a low stable rank that is well below the linear algebraic rank of the subspace, degrading fine-tuning performance. To mitigate the underutilization of the allocated subspace, we propose PoLAR, a parameterization inspired by the polar decomposition that factorizes the low-rank update into two direction matrices constrained to Stiefel manifolds and an unconstrained scale matrix. Our theory shows that PoLAR yields an exponentially faster convergence rate on a canonical low-rank adaptation problem. Pairing the parameterization with Riemannian optimization leads to consistent gains on three different benchmarks testing general language understanding, commonsense reasoning, and mathematical problem solving with base model sizes ranging from 350M to 27B.

We study compute efficiency of LLM training when using different parameterizations, i.e., rules for adjusting model and optimizer hyperparameters (HPs) as model size changes. Some parameterizations fail to transfer optimal base HPs (such as learning rate) across changes in model depth, requiring practitioners to either re-tune these HPs as they scale up (expensive), or accept sub-optimal training when re-tuning is prohibitive. Even when they achieve HP transfer, we develop theory to show parameterizations may still exist in the lazy learning regime where layers learn only features close to their linearization, preventing effective use of depth and nonlinearity. Finally, we identify and adopt the parameterization we call CompleteP that achieves both depth-wise HP transfer and non-lazy learning in all layers. CompleteP enables a wider range of model width/depth ratios to remain compute-efficient, unlocking shapes better suited for different hardware settings and operational contexts. Moreover, CompleteP enables 12-34% compute efficiency improvements over the prior state-of-the-art. All experiments were run on Cerebras CS-3 systems.

Approaches for goal-conditioned reinforcement learning (GCRL) often use learned state representations to extract goal-reaching policies. Two frameworks for representation structure have yielded particularly effective GCRL algorithms: (1), in which methods learn successor features with a contrastive objective that performs inference over future outcomes, and (2), which link the (quasimetric) distance in representation space to the transit time from states to goals. We propose an approach that unifies these two frameworks, using the structure of a quasimetric representation space (triangle inequality) with the right additional constraints to learn successor representations that enable optimal goal-reaching.

While parameter-efficient fine-tuning methods like low-rank adaptation (LoRA) are standard for large language models, principled estimation of epistemic uncertainty remains challenging. Recent results in the LoRA regime suggest that discrete multi-mode approaches such as deep ensembles offer little benefit over single-mode methods. This contradicts broader observations in deep learning, where ensembling independent optima typically improves generalization, and linking these modes through continuous low-loss valleys further enhances Bayesian model averaging (BMA). Whether such structure exists in the LoRA space and whether it yields functional diversity missed by local or discrete methods has not been studied. We introduce LoRA-Curve, a segmented Bézier curve parameterization in the LoRA space, with two variants: a free configuration that jointly optimizes all control points, and an anchored configuration that connects independently fine-tuned LoRA optima. We prove pathwise continuity and Lipschitz regularity of the loss along the curve and empirically show, across reasoning and classification benchmarks with Qwen2.5 7B, that linear interpolation encounters loss barriers, while our anchored multi-segment curves connect independent optima through continuous low-loss valleys. Combined with flat-minima perturbations and a Jensen-Shannon divergence regularizer, LoRA-Curve yields measurably higher mutual information of the predictive distribution without sacrificing performance, and links continuous parameter-space traversal to functional diversity.