A Survey of Weight Space Learning: Understanding, Representation, and Generation
Authors: Xiaolong Han, Zehong Wang, Bo Zhao, Binchi Zhang, Jundong Li, Damian Borth, Rose Yu, Haggai Maron, +3 more
Organizations: University of Surrey · University of Notre Dame · University of California San Diego · University of Virginia · University of St.Gallen · Technion · Nvidia
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.
Figures & tables
Figure 1: Overview of the survey. We organize weight space learning into three dimensions: understanding weight-space properties, learning compact representations, generating weights via auxiliary models. In addition, it consolidates applications to related domains and benchmarks, paving the way for future research.
Figure 2: Conceptual overview of weight space learning. (a) Weight space understanding analyzes symmetries in weight space and introduces quotient spaces to reduce redundancy and clarify functional equivalence. (b) Weight space representation learns compact embeddings of neural weights, discriminating symmetry in weight space well. (c) Weight space generation synthesizes network weights, supporting efficient model creation and adaptation.
Figure 4: Weight space symmetry and its induced functional manifestations. (a) illustrates the invariance under group action: applying a transformation ρin to weights θ in weight space does not alter the induced function in quotient space 2 2 2 The quotient space is defined as the set of equivalence classes of weights that parameterize the same function. Mathematically, let G be the symmetry group acting on the weight space W . The quotient space is defined as the orbit space W/G:={[θ]∣θ∈W} , where the equivalence class [θ]={g⋅θ∣g∈G} contains all weight configurations reachable via symmetry transformations. . (b) illustrates the equivariance under group action: transforming weights θ by ρin in weight space induces a corresponding transformation ρout in quotient space.
Functional Relation
Transformation Example
Functional Effect
Neuron permutation
Functionally identical network under reordered hidden units
Invariance
Positive scaling
Scaling is neutralized in batch normalization
Bias translation in Softmax
Constant shift of logits leaves probabilities unchanged
Orthogonal rotation
Rotating neuron basis with output changing accordingly
Equivariance
Sign flipping
Output changes predictably under the corresponding sign transformation
Table 1: Illustrative examples of functional invariance and equivariance induced by weight space symmetry. Under weight space symmetry transformations, invariance means that the realized function remains unchanged, while equivariance means that the realized function changes in a predictable, structured manner. These relations explain that many different parameter configurations correspond to the same or systematically related functions. See Zhao et al. [2025b] for a theoretical discussion.
Figure 5: Weight Space Representation Methods: Overview and Comparison. (a) Model-based approaches. All take raw weight space θ as input but differ in how they handle symmetry. Symmetry-agnostic methods learn representations directly from weight distributions without explicitly encoding neuron permutation symmetry. Symmetry-aware methods incorporate dedicated equivariant modules to respect permutation invariance. Graph-based symmetry approaches leverage GNNs to automatically capture structural symmetries among neurons. (b) Model-free approaches. Probe-based behavioral learning infers representations from functional responses to designed probes, enabling embeddings that encode behavioral properties and functional symmetries of the underlying network.
Figure 6: Weight Space Generation Methods: Overview and Comparison. (a) Hypernetworks. A conditioning prompt is processed by a hypernetwork H(⋅) to generate the parameters θ of a target model f(⋅) . The hypernetwork is trained end-to-end using the downstream task loss Ltask(fθ) , enabling fast and conditional weight generation. (b) Generative models. Taking a diffusion-based approach as an example, the process begins by sampling from a noise distribution in weight space. Through iterative denoising steps, the model progressively refines the noisy initialization toward a structured representation of network parameters. This forward–reverse diffusion mechanism learns the underlying distribution of weights and enables conditional generation, producing diverse and task-aligned neural network configurations without explicit gradient-based optimization.
Method
Conditioning
Diversity
Data Source
Suitable Scenarios
Hypernetworks
✓
×
real-world data
Few-shot and fast adaptation
Generative models
P
✓
model zoo
Large-scale and high-fidelity generation
Table 2: Comparison of Hypernetworks and generative models. ✓ indicates full satisfaction, × indicates the condition is not satisfied, and P denotes partial satisfaction.
Domain
Representative References
Why WSL Fits
Implicit Neural Representations (INR)
Shamsian et al. [2023] ; Shamsian et al. [2024] ; Huang et al. [2025] ; Zhou et al. [2023b] ; inr2vec [ Luigi et al., 2023 ] ; Functa [ Dupont et al., 2022a ] ; Spatial Functa [ Bauer et al., 2023 ] ; NeoMLP [ Kofinas et al., 2024b ] ; GRAF Schwarz et al. [2020] ; pi-GAN Chan et al. [2021] ; HyperDiffusion Erkoç et al. [2023] ; Yang and Wang [2024] ; Gielisse and van Gemert [2025] .
Data can be naturally represented by weights.
Model Unification
GEM [ Du et al., 2021 ] ; Functa [ Dupont et al., 2022a ] ; Spatial Functa [ Bauer et al., 2023 ] ; uGNN [ Pala and Rekik, 2025 ] .
Architecture-agnostic model space
Continual Learning
von Oswald et al. [2020] ; W-Diff [ Xie et al., 2024 ] ; Li et al. [2025] .
Forgetting mitigation via weight regeneration
Meta Learning
MetaDiff [ Zhang et al., 2024a ] ; Mc-Di [ Guan et al., 2025 ] ; MH [ Zhao et al., 2020 ] .
Fast gradient-free adaptation
Federated Learning
FedDiff [ Li et al., 2024a ] ; pFedHN [ Shamsian et al., 2021 ] ; pFedGPA [ Lai et al., 2025 ] .
Personalized weights, low communication
Neural Architecture Search
GHN [ Zhang et al., 2019 ] ; GHN-1/2 [ Knyazev et al., 2021 ] .
Efficient weights prediction
Table 3: Applications of Weight space learning to Related Domains.