Weight-Space Learning

Weight-space learning across architectures requires representations that respect parameter symmetries and scale to large networks. Existing equivariant tensor encoders require architecture-specific symmetry specifications, while neuron graphs make global attention costly. We introduce a typed graph organized around shared coordinate-permutation actions. Its nodes, called interfaces, represent coordinate spaces whose bound tensor axes must be permuted jointly; operation occurrences, called modules, form typed hyperedges. A compiler derives this graph from supported PyTorch models, with exact recovery of the permutation factors defined by its constraints. The Universal Weight Encoder (UWE) combines local equivariant updates to parameter features with global attention over invariant interface summaries. The small number of interfaces makes global attention practical, and a single encoder processes different supported architectures. UWE is competitive with architecture-specific state-of-the-art methods on established benchmarks. On a new zoo of 1,200 transformers spanning 100 architectures, it outperforms the evaluated baselines at ranking networks within unseen architectures, including architectures larger than those used for training, while using substantially less memory.

Weight-space learning treats the weights of neural networks as data, processed by other models called metanetworks. A key challenge is that different weights can realize the same function, related by a group of symmetries the metanetwork should respect. While symmetry-aware metanetworks for simple architectures such as MLPs are well developed, a comprehensive treatment of transformers, and in particular transformer-based language models, is still missing. Importantly, their weight space has a richer symmetry group: beyond neuron permutations, attention introduces continuous general-linear symmetries of the attention matrices and permutations of entire heads. Existing methods for transformer weights either capture only part of these symmetries or are restricted to a single, fixed architecture, and none accounts for language-model-specific components such as the token embeddings. Building on graph metanetworks, we propose a graph-based approach that is both symmetry-aware and applicable across depths and widths. We formalize a weight-space symmetry group of complete decoder-only language models and provide a graph construction that is invariant to it. We further introduce two new transformer-population datasets with novel tasks, such as training-corpus prediction and fine-tuning detection, on which our approach notably outperforms existing baselines.

Learning properties of neural networks has recently attracted growing interest, with existing approaches operating either directly on network parameters or through probe-based representations of network behavior. While probing methods have shown strong empirical performance, their theoretical foundations remain limited. In this work, we study when finite probe-based representations are sufficient for learning neural functionals. We establish general identification and universality results for probing, and show that using intermediate hidden representations can provide significantly more informative representations than relying only on final outputs. Motivated by these results, we introduce HIDDENPROBE, a simple architecture for learning from hidden probe responses. Across a range of neural functional benchmarks, including both MLPs and Transformers, HIDDENPROBE consistently improves over existing probing methods and achieves state-of-the-art performance. Our code is publicly available on GitHub.

Implicit Neural Representations (INRs) encode signals as the weights of a coordinate-based neural network and have recently been proposed as an alternative domain for downstream learning. While promising, classification directly in weight space remains challenging due to the high dimensionality and complex structure of INR parameters. Furthermore, the way discriminative information is distributed across INR weights remains poorly understood. We propose a hierarchical Mixture-of-Experts (HMoE) Transformer that processes INR weights using conditional computation aligned with the structure of the underlying implicit network. Coupled with a meta-learning framework that shapes INR parameters for downstream tasks, our model achieves state-of-the-art accuracy across standard benchmarks, ranging from low-resolution datasets to high-resolution ImageNet-1K. To gain insight into how INRs encode discriminative information, we develop weight-space attribution and pruning methods that identify parameters most relevant for classification. These analyses reveal how class-specific structure emerges within INR layers and support the suitability of MoE architectures for weight-space learning. Our approach advances both the performance and interpretability of weight-space classifiers.

Many neural networks operations have a multiplicative nature rather than additive: halving or doubling a norm are analogous relatively but require unequal optimization distances when taking linear steps. Adaptive optimizers such as Adam normalize updates per coordinate, but update steps remain additive; weights with very different magnitudes receive similarly sized absolute changes, producing very different relative perturbations. We introduce \method (\methodshort), a weight reparameterization for neural networks that combines a sign-aware symmetric-exponential pathway with an identity-like linear pathway. The symmetric-exponential pathway is near-linear for small raw weights but increasingly curved at larger magnitudes. Additive updates in logarithmic space map to magnitude-proportional changes in effective weight space. The linear pathway provides a direct route through the transform that we hypothesize stabilizes optimization, while learnable scale, curvature, and offset parameters control balance between pathways and the curvature of the exponential pathway. These components create a curved parameter-space geometry that empirically improves speed of loss descent over standard linear parameterization. We also identify a useful mismatched initialization: raw weights are chosen so a symmetric version of the transform matches Xavier statistics, but training uses an asymmetric forward transform that leaves positive weights at full strength while making negative weights smaller in magnitude; in small-model ablations, this improves early optimization and may act as a form of symmetry breaking. We train transformers on OpenWebText over nine width$\times$depth configurations, \methodshort reaches matched validation loss in 1.32--1.49$\times$ fewer training steps, with the largest widths seeing the biggest gains.

Weight space learning aims to learn representations of neural network (NN) weights, enabling different downstream tasks. Existing approaches show promising performance, but lacking a way to shape these weight-space representations using information about the datasets the models were trained on, thus limiting downstream applications. We propose WeightCLIP, a method for learning a dataset-aligned latent space for neural networks, where datasets information is induced during training. The NNs are encoded as latent representations using an autoencoder, while dataset samples are encoded using a dataset encoder. The two representations are aligned using a contrastive objective, effectively reshaping the weight-space representations according to the datasets. We demonstrate that such representations can be used for different downstream tasks, including mapping dataset information to a weight-space representation that decode to strong models. In addition, we introduce a latent refinement process for generating models that outperforms standard fine-tuning. Overall, our results demonstrate that explicitly incorporating dataset information improves what can be achieved with weight-space representations across retrieval, generation, and refinement. Code will be available at https://github.com/HSG-AIML/WeightCLIP.

The rapid advancements in using neural networks as implicit data representations have attracted significant interest in developing machine learning methods that analyze and process the weight spaces of other neural networks. However, efficiently handling these highdimensional weight spaces remains challenging. Existing methods often overlook the sequential nature of layer-by-layer processing in neural network inference. In this work, we propose a novel approach using dynamic graphs to represent neural network parameters, capturing the temporal dynamics of inference. Our Dynamic Neural Graph Encoder (DNG-Encoder) processes these graphs, preserving the sequential nature of neural processing. Additionally, we also leverage DNG-Encoder to develop INR2JLS (Implicit Neural Representation to Joint Latent Space) for facilitate downstream applications, such as classifying Implicit Neural Representations (INRs). Our approach demonstrates significant improvements across multiple tasks, surpassing the state-of-the-art INR classification accuracy by approximately 10% on the CIFAR-100-INR.

Offline reinforcement-learning losses (RFT, RIFT, DFT, Offline GRPO, DPO) are widely used to distill reasoning from large teachers into smaller students, and are typically compared on downstream accuracy alone. We ask whether they are mechanistically distinct or converge to a similar weight update. Training six methods (SFT, RFT, DFT, RIFT, Offline GRPO, DPO) on identical math rollouts from a single base model (Qwen3-4B) with attention-only LoRA, we analyze the resulting deltas via cosine similarity, principal-angle subspace analysis, linear mode connectivity, and CKA. We observe: (i) SFT, RFT, and RIFT have nearly colinear weight deltas (cosine >= 0.97, top-1 principal angle ~7 deg median over 144 modules) and comparable GSM8K accuracy (87-88%, n=1319; pairwise McNemar p >= 0.15); (ii) DFT diverges further in direction than any reward-weighted method despite using the same data; (iii) Offline GRPO adds a substantial component orthogonal to the SFT direction (~67% globally, up to ~86% in late layers) while staying in the SFT loss basin; (iv) DPO sits in a near-orthogonal subspace, shows a mode-connectivity barrier, and collapses late-layer CKA to ~0.46. DPO also reaches the highest accuracy in our protocol on both GSM8K (93.5%, McNemar p < 10^-9 vs. each other method) and AIME26 (30.0% vs. 3.3-10.0%); its training uses a 10x smaller learning rate than the others (the standard convention), so the update-norm and accuracy gaps reflect loss-function and optimizer choices jointly, and a learning-rate-matched DPO comparison is left for future work.

The explosive growth of open-source model repositories has created a Model Jungle, where checkpoints are frequently shared without adequate documentation or metadata. While weight-space learning offers a pathway to identify and analyze these models directly from their parameters, processing full-scale weights is computationally prohibitive. Probing-based methods have emerged as a lightweight alternative, extracting permutation-equivariant representations via learnable probe vectors. However, existing probing methods are limited by a single-view design: they capture first-order structures but fail to encode the rich, higher-order correlation patterns inherent in row-column interactions. To bridge this gap, we introduce MVProbe, a multi-perspective probing framework that synthesizes first-order signals with interaction-aware (Gram-based) views. Our approach is theoretically grounded; we analyze the scaling laws of different probing orders to derive a principled standardization and fusion strategy that ensures balanced contributions from all branches. On the Model Jungle benchmark, MVProbe consistently outperforms the state-of-the-art ProbeX across diverse architectures, including discriminative backbones (ResNet, SupViT, MAE, DINO) and large-scale generative LoRA adapters (Stable Diffusion LoRA).

Neural network checkpoints have quietly become a large-scale data resource: millions of trained weight vectors now exist, each encoding task-, domain-, and architecture-specific knowledge. This position paper argues that model checkpoints should be treated as a first-class data modality, and that generative modeling in weight space should be standardized as a core machine learning primitive. Recent advances demonstrate that neural weights can be synthesized on demand, often matching fine-tuning performance while reducing adaptation cost by orders of magnitude. We contend that these results reflect an underlying structural fact: high-performing models occupy low-dimensional, highly structured regions of weight space shaped by symmetry, flatness, modularity, and shared subspaces. Building on this view, we organize existing methods into a five-stage pipeline, survey applications where the approach is already practical, and clarify current limits: adapter-scale and conditional generation are advancing rapidly, while unrestricted frontier-scale checkpoint synthesis remains open. Our goal is to shift the community's default mindset from optimizing models per task to sampling models from learned weight distributions, accelerating toward an era in which AI systems routinely improve or create other AI systems.

Metanetworks are neural architectures designed to operate directly on pretrained weights to perform downstream tasks. However, the parameter space serves only as a proxy for the underlying function class, and the parameter-function mapping is inherently non-injective: distinct parameter configurations may yield identical input-output behaviors. As a result, metanetworks that rely solely on raw parameters risk overlooking the intrinsic symmetries of the architecture. Reasoning about functional identity is therefore essential for effective metanetwork design, motivating the development of equivariant metanetworks, which incorporate equivariance principles to respect architectural symmetries. Existing approaches, however, typically enforce strict equivariance, which imposes rigid constraints and often leads to sparse and less expressive models. To address this limitation, we introduce the novel concept of quasi-equivariance, which allows metanetworks to move beyond the rigidity of strict equivariance while still preserving functional identity. We lay down a principled basis for this framework and demonstrate its broad applicability across diverse neural architectures, including feedforward, convolutional, and transformer networks. Through empirical evaluation, we show that quasi-equivariant metanetworks achieve good trade-offs between symmetry preservation and representational expressivity. These findings advance the theoretical understanding of weight-space learning and provide a principled foundation for the design of more expressive and functionally robust metanetworks.

Recent advances in remote sensing have led to an increase in the number of available foundation models; each trained on different modalities, datasets, and objectives, yet capturing only part of the vast geospatial knowledge landscape. While these models show strong results within their respective domains, their capabilities remain complementary rather than unified. Therefore, instead of choosing one model over another, we aim to combine their strengths into a single shared representation. We introduce GeoSANE, a geospatial model foundry that learns a unified neural representation from the weights of existing foundation models and task-specific models, able to generate novel neural networks weights on-demand. Given a target architecture, GeoSANE generates weights ready for finetuning for classification, segmentation, and detection tasks across multiple modalities. Models generated by GeoSANE consistently outperform their counterparts trained from scratch, match or surpass state-of-the-art remote sensing foundation models, and outperform models obtained through pruning or knowledge distillation when generating lightweight networks. Evaluations across ten diverse datasets and on GEO-Bench confirm its strong generalization capabilities. By shifting from pre-training to weight generation, GeoSANE introduces a new framework for unifying and transferring geospatial knowledge across models and tasks. Code is available at hsg-aiml.github.io/GeoSANE/.

Each LoRA checkpoint compactly stores task-specific updates in low-rank weight matrices, offering an efficient way to adapt large language models to new tasks and domains. In principle, these weights already encode what the adapter does and how well it performs. In this paper, we ask whether this information can be read directly from the weights, without running the base model or accessing training data. A key obstacle is that a single LoRA update can be factorized in infinitely many ways. Without resolving this ambiguity, models trained on the factors may fit the particular factorization rather than the underlying update. To this end, we propose \methodfull, which maps each LoRA update to a provably canonical form via QR decomposition followed by SVD, so that all equivalent factorizations share the same representation. The resulting components are then tokenized and processed by a Transformer to produce a weight-space embedding. Across language and vision LoRA collections, W2T achieves strong results on attribute classification, performance prediction, and adapter retrieval, demonstrating that LoRA weights reliably indicate model behavior once factorization ambiguity is removed. Code is available at https://github.com/xiaolonghan2000/Weight2Token.

Neural network weights are typically viewed as the end product of training, while most deep learning research focuses on data, features, and architectures. However, recent advances show that the set of all possible weight values (weight space) itself contains rich structure: pretrained models form organized distributions, exhibit symmetries, and can be embedded, compared, or even generated. Understanding such structures has tremendous impact on how neural networks are analyzed and compared, and on how knowledge is transferred across models, beyond individual training instances. This emerging research direction, which we refer to as Weight Space Learning (WSL), treats neural weights as a meaningful domain for analysis and modeling. This survey provides the first unified taxonomy of WSL. We categorize existing methods into three core dimensions: Weight Space Understanding (WSU), which studies the geometry and symmetries of weights; Weight Space Representation (WSR), which learns embeddings over model weights; and Weight Space Generation (WSG), which synthesizes new weights through hypernetworks or generative models. We further show how these developments enable practical applications, including model retrieval, continual and federated learning, neural architecture search, and data-free reconstruction. By consolidating fragmented progress under a coherent framework, this survey highlights weight space as a learnable, structured domain with growing impact across model analysis, transferring, and weight generation. We release an accompanying resource at https://github.com/Zehong-Wang/Awesome-Weight-Space-Learning.

The escalating parameter counts in modern deep learning models pose a fundamental challenge to efficient training and resolution of overfitting. We address this by introducing the Mapping Networks which replace the high dimensional weight space by a compact, trainable latent vector based on the hypothesis that the trained parameters of large networks reside on smooth, low-dimensional manifolds. Henceforth, the Mapping Theorem enforced by a dedicated Mapping Loss, shows the existence of a mapping from this latent space to the target weight space both theoretically and in practice. Mapping Networks significantly reduce overfitting and achieve comparable to better performance than target network across complex vision and sequence tasks, including Image Classification, Deepfake Detection etc, with $\mathbf{99.5\%}$, i.e., around $500\times$ reduction in trainable parameters.

Weight-space models learn directly from the parameters of neural networks, enabling tasks such as predicting their accuracy on new datasets. Naive methods -- like applying MLPs to flattened parameters -- perform poorly, making the design of better weight-space architectures a central challenge. While prior work leveraged permutation symmetries in standard networks to guide such designs, no analogous analysis or tailored architecture yet exists for Kolmogorov-Arnold Networks (KANs). In this work, we show that KANs share the same permutation symmetries as MLPs, and propose the KAN-graph, a graph representation of their computation. Building on this, we develop WS-KAN, the first weight-space architecture that learns on KANs, which naturally accounts for their symmetry. We analyze WS-KAN's expressive power, showing it can replicate an input KAN's forward pass - a standard approach for assessing expressiveness in weight-space architectures. We construct a comprehensive ``zoo'' of trained KANs spanning diverse tasks, which we use as benchmarks to empirically evaluate WS-KAN. Across all tasks, WS-KAN consistently outperforms structure-agnostic baselines, often by a substantial margin. Our code is available at https://github.com/BarSGuy/KAN-Graph-Metanetwork.

We investigate the potential of weights to serve as effective representations, focusing on neural fields. Our key insight is that constraining the optimization space through a pre-trained base model and low-rank adaptation (LoRA) can induce structure in weight space. Across reconstruction, generation, and analysis tasks on 2D and 3D data, we find that multiplicative LoRA weights achieve high representation quality while exhibiting distinctiveness and semantic structure. When used with latent diffusion models, multiplicative LoRA weights enable higher-quality generation than existing weight-space methods.