Weight-Space

Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces. We develop a statistical-mechanical description of learning directly in function space, treating parameter configurations as microscopic realizations and functions with their dynamical operators as macroscopic variables. For mean-squared loss, the exact error dynamics are governed by the learning operator $M=JJ^\ast$. Combining the dynamical Boltzmann weight of the conditional stochastic dynamics with the parameter-space density of states, whose local curvature defines a statistical operator $B$, and integrating over local fluctuations yields $\Phi_{\mathrm{fluc}}(M;B)=\frac{\sigma_\xi^2}{2}\log\det(M^{-1}+B)+\mathrm{const}.$ At fixed spectrum, this term is rotationally stationary when $[M,B]=0$, is minimized by pairing large eigenvalues of $M$ with small eigenvalues of $B$, and generates a local restoring contribution against rotational mismatch. For ReLU-type function spaces under mild stable statistical conditions, $B=\sigma_\xi^2L^\ast\mathcal K L$, where $L$ measures coarse-grained second-order structure. Thus the low-$B$ sector corresponds, up to bounded anisotropy of $\mathcal K$, to low structural curvature, implying a preference for faster relaxation along smooth, data-adaptive directions. These results identify function space as a natural macroscopic level for studying stable collective organization in learning.

Task arithmetic composes skills by adding weight displacements, and merged models are then judged on benchmark suites. We measure when that composition is functionally additive, and find that the answer depends as much on how the model is prompted as on which tasks are merged. Across two-dimensional composition surfaces -- five model settings from 0.5B to 8B, two families, LoRA and full fine-tuning -- pairwise non-additivity is real, seed-stable, and transfers in coarse order to unseen task pairs: all eight preregistered sign predictions held. But it is input-conditioned everywhere we measured: the same merged model that shows a six-point interaction contrast on code prompts shows none on math prompts, and wrapping the identical code prompts in the instruction template the adapters were trained on collapses the contrast twenty-fold, from +6.9 to +0.3 points -- while re-serializing them in an untrained chat template leaves it intact (+12.5), falsifying our own preregistered prediction. Execution benchmarks (pass@1) inherit the training-format wrapper's blindness. Weight-space composition therefore supports coarse, input- and format-conditioned functional statements -- not a universal merging-performance predictor, and not one that training-format evaluations can see.

Open-weight large language models (LLMs) are increasingly developed through complex, multi-stage pipelines, leading to intricate lineage relationships that reflect model origin, ownership, and evolution. Understanding these relationships is important for model provenance, governance, and supply-chain integrity. In this work, we investigate the notion of LLM "biometrics" (analogous to human biometrics) to ask whether LLMs exhibit intrinsic fingerprints in weight space alone, without access to input data, that reveal their origin and lineage. We formulate this as a lineage discrimination problem, distinguishing among independent-origin, same-series, and shared-base models. To characterize these relationships, we propose a unified geometric fingerprinting framework that analyzes weight matrices from two complementary perspectives: (i) spectral energy, captured by singular value distributions to encode global magnitude patterns, and (ii) subspace alignment, quantified via subspace deviations to capture directional geometry. Our analysis uncovers a clear hierarchy of structural similarity in weight space: spectral energy reliably distinguishes independently trained models and different model families, while subspace alignment enables fine-grained discrimination among closely related models, including variations in dataset scale and post-training procedures. Extensive experiments on over 110 diverse open-weight LLM pairs demonstrate that weight-space geometry provides a robust and interpretable signal for model lineage, enabling coarse-grained regime separation and fine-grained discrimination within shared-base models.

Vector symbolic architectures (VSA) are widely used for reasoning in neuro-symbolic (NeSy) AI, yet high-dimensional codebooks often create severe memory bottlenecks that limit scalability and deployment. In this paper, we propose Gram-Space, a compression framework that applies Gram-Schmidt orthogonalization to represent codebook vectors in a compact orthonormal coordinate system. Gram-Space preserves the dot-product structure required by matrix-based VSA operators, which supports numerically equivalent execution of matrix similarity, probability vectorization, and attention score computations. We provide a correctness analysis showing that inner products are preserved under the orthonormal basis representation. Using modern GPU hardware, we benchmark the Gram-Space framework on standard neuro-symbolic reasoning datasets. Experimental evaluations across state-of-the-art VSA models show that Gram-Space reduces model-level GPU memory usage by up to 15.75x and improves inference latency by up to 3.62x. Profiling results further indicate that Gram-Space reduces allocation-heavy overhead in codebook-associated stages and improves hardware utilization for NeSy workloads.

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 \textbf{\method} (\textbf{\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 \emph{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.

Large language models exhibit recurring behavioural patterns -- personas -- that shape generalisation and safety, but we lack reliable tools for decomposing, measuring, and controlling them. Our central insight is to treat personas as positions in a space of behavioural traits, using the OCEAN framework to describe model personas in terms of Openness, Conscientiousness, Extraversion, Agreeableness, and Neuroticism. We train low-rank adapters to amplify or suppress individual traits, and evaluate their effects using an LLM-judge calibrated against a human-validated panel, trait-specific multiple-choice benchmarks, and standard capability evaluations. Across six models from three families (4B-32B), we find that each adapter moves its target trait largely monotonically with scale, combines approximately additively with other adapters to construct mixed personas, and preserves performance on capability benchmarks at moderate scales. We further show that the induced trait axes affect safety-relevant behaviour in downstream evaluations: for example, moving along neuroticism and agreeableness axes affects frustration and sycophancy respectively. We also introduce an unsupervised psychometric pipeline that recovers four interpretable behavioural factors (tone, initiative, didacticism, epistemic caution) from model rollouts. Persona control can then be considered in terms of learning, scaling, and composing traits in weight space, providing a bridge between personality measurement, model editing, and safety.

Lattice field theory is the workhorse of non-perturbative physics, used to simulate phenomena from the strong nuclear force to critical phenomena in materials. Its Boltzmann distributions are parametrized analytically by coupling constants, but these bare parameters are weak predictors of observables -- extracting physics typically requires extensive simulation. While normalizing flows have emerged as effective samplers at fixed couplings, it remains difficult to interpret what these networks have learned. This raises a natural question: can the physics be read off directly from the flow network parameters themselves, and can those parameters be generated for unseen theories? We propose lattice field theory as a testbed for neural network interpretability: because the target physics is qualitatively well-understood and smoothly varying, it provides ideal synthetic data with known ground truth. To this end, we introduce JEPAWG, a Joint-Embedding Predictive Architecture-based Weight Generator that maps couplings directly to flow weights via a learned latent space. On a scalar theory at lattices of size $6^2$ to $11^2$, the JEPAWG latent space recovers the correct intrinsic dimension of the underlying manifold, locates the phase transition, and encodes a finite-size shift aligned with the 2D Ising exponent $ν\approx 1$, allowing us to uncover physical structure by studying the network weights alone. This suggests the fascinating idea of treating the network weights as a new type of physical observable. As a generator, JEPAWG also interpolates and extrapolates to unseen couplings effectively and remains robust to weight-space discontinuities introduced by multi-seed training data, outperforming PCA, AE, and VAE baselines.

A persistent empirical observation is that trained neural networks outperform their neural tangent kernel (NTK) limit on tasks with compositional structure, yet a quantitative account of $\textbf{when}$ and $\textbf{by how much}$ has been lacking. Working on the unit circle, we give such an account through a dichotomy between two complexity measures of the target: its $\textbf{Fourier complexity}$, which controls NTK kernel regression, and its $\textbf{architectural complexity}$, which controls learning over depth-$L$, width-$w$ ReLU networks with the variation norm of the weights bounded by $R$. We first characterize the minimax rate of the architecture class $\mathcal{C}_{L,w,R}$, pinning it down up to a single factor of $L$: between $Ω(Lw^2R^2/n)$ and $\tilde{O}(L^2w^2R^2/n)$. We then show the NTK estimator sits $\textbf{exponentially}$ above this floor whenever the two complexities decouple: for the depth-$L$ iterated sawtooth, NTK regression needs $Ω(4^L)$ samples while the minimax floor is polynomial in $L$. Numerical experiments confirm the theoretical claims: on bandlimited smooth targets, the NTK is competitive or better, while on the hypercube sparse-parity model, a standard two-layer network beats the NTK by four to six orders of magnitude in test error. The gap is thus a function-space property, a mismatch between the kernel's smoothness bias and the target's compositional structure, rather than a generic kernel-versus-network phenomenon.

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.

Foundation models are routinely released to the public, yet the data recipes used to train them -- such as domain mixture weights that determine how different sources are sampled -- are rarely disclosed. This creates an access asymmetry: researchers study the resulting models but lack visibility into the training distribution that produces them. Prior works for inferring training data, such as membership inference, detect at the level of individual samples and thus cannot characterize the global composition of the training corpus. We introduce WARP, a framework that recovers a fine-tuned model's training mixtures directly from its released weights. WARP interpolates between the base and fine-tuned models using model merging, generating pseudo-checkpoints that approximate the missing training trajectory and expose a geometric footprint of the training data in the weight space. From these simulated footprints, WARP extracts geometric features and maps them to domain proportions using either a parameter-free softmax readout or an MLP projector trained on synthetic mixtures. In controlled experiments with BERT and GPT-2, WARP recovers domain mixtures with an average MAE as low as 0.046 and 0.104 respectively, outperforming membership inference and a variant with access to the true training trajectory.

Full fine-tuning adapts large language models to new tasks but can erode capabilities they already possess. Existing remedies protect through proxies such as parameter distances, importance penalties, output matching, or dominant singular directions of the weights, but none directly asks which activation directions the preserved capability relies on. We argue that a capability is characterized more faithfully by the activation subspace it induces than by the singular geometry of the weight matrix, and develop function-space protection, instantiated as FORA (Function-space Orthogonal Residual Adaptation). From label-free calibration inputs, FORA estimates, per layer, the principal directions $Q$ of the input-activation covariance and forms a right projector $P_Q = I - QQ^T$. Paired with a left projector $P_U$ from the weight SVD, the update is $ΔW = P_U M P_Q + U_2 D_δ V_2^T$: a high-capacity branch structurally barred from reading capability-relevant function directions, plus a narrow spectral channel for controlled plasticity. The construction extends to parameter-efficient adaptation via $M \to (α/r) BA$. Across three settings on Qwen3-1.7B, including COGS and GSM8K learned while preserving translation and translation learned while preserving math, FORA consistently improves preservation over weight-space projection and standard regularization, with only a small new-task trade-off in the math-preservation setting. A controlled ablation isolating the projection source shows that the advantage comes not from projection itself, but from projecting onto capability-derived rather than weight-derived directions. Code is available at https://github.com/zrui239/FORA.

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.

Principled regression for stochastic processes is a long-standing challenge with deep connections to scientific inverse problems. We introduce Flow Annealing Posterior Sampling (FAPS), to our knowledge the first function-space posterior sampling framework that unifies stochastic-process regression and PDE inverse problems. Built on pretrained function-space flow-matching priors, FAPS enables likelihood-guided posterior inference from sparse and noisy observations, supports variable query discretizations, and avoids explicit prior-density evaluation. Its Langevin correction uses a low-rank covariance preconditioner to exploit dominant function-space correlations across discretizations. Across Gaussian and non-Gaussian stochastic-process regression benchmarks and diverse PDE inverse problems, FAPS produces coherent posterior samples with accurate uncertainty quantification, significantly outperforming existing functional regression baselines and achieving competitive or better PDE noisy inverse performance than diffusion-based posterior samplers while reducing test-time sampling cost.

Meta-reinforcement learning is a promising approach to multi-objective optimisation because it enables rapid policy adaptation across changing environments and preference settings. However, conventional few-shot methods usually fine-tune from a single shared meta-policy, which can reduce solution diversity and limit exploration of the Pareto front, especially in high-dimensional combinatorial problems such as supply chain optimisation. We propose a population-based Meta-reinforcement learning framework that combines decomposition with evolutionary search in scalarisation weight space. The framework maintains a population of weight vectors, each associated with a distinct meta-policy trained through gradient-based meta-learning, and iteratively refines this population through elitist selection, crossover, and mutation guided by hypervolume and entropy contributions. We evaluate the method in a multi-objective supply chain setting with conflicting economic, environmental, and social goals, and further test its generality on standard reinforcement learning problems. The results show that the proposed approach yields more diverse, better distributed Pareto front approximations, improves cross-task adaptation, increases hypervolume by up to 32% over Meta-multi-objective reinforcement learning in the complex case, and attains the lowest average Hausdorff distance among all compared methods.

Parameter decomposition (PD) decomposes neural networks into interpretable computational components that faithfully reflect the original network's operations. However, scaling PD to large models requires vast compute, making it a costly and risky endeavor. Here we propose targeted PD (tPD), which identifies only the components that process specific inputs of interest -- from isolated prompts to large subtasks -- by introducing a high-rank catch-all component that handles all non-target data. We validate tPD on toy models and on transformer language models trained on The Pile, where it recovers reproducible, mechanistically faithful circuits. We extract a CSS-only submodel of a 4-block transformer using 7% of the FLOPs of its published decomposition, and in a 12-block transformer we surgically ablate and rewire memorized sequences, with negligible side effects on other inputs.

Recurrent networks can contain substantial functional redundancy in weight space: changing a recurrent matrix may leave the input-output rollout nearly unchanged on a task distribution, while similar-scale changes can destroy the same behavior. We study this redundancy in one-layer tanh RNNs using ordered real Schur coordinates. The Schur form separates spectral blocks from directed nonnormal couplings, giving a diagnostic basis for structured ablations that keep the input and readout maps fixed. In a fixed-length copy task, selected nonnormal Schur couplings can be removed with little loss in some trained solutions, whereas other couplings are necessary for accurate autonomous replay. Across flip-flop, sine generation, and context-dependent integration, the loss-preserving ablation profile varies across tasks and trained solutions. These results identify candidate approximate functional invariances, not universal symmetries of recurrent weight space. Schur-coordinate ablations provide a practical diagnostic for which structured perturbations preserve a trained recurrent solution and which ones disrupt its computation.

Catastrophic forgetting in continual adaptation is usually studied through parameter drift, replay, or distillation, but these views do not identify which output-space directions are vulnerable. We give a function-space account in the NTK regime: new-task training induces old-task prediction drift through the cross-task kernel, yielding a closed-form predictor for the forgetting vector before any new-task gradient step. In frozen-backbone linear-head PEFT-CL, where the model is linear in the trainable parameters, the predictor is exact up to numerical precision; for nonlinear adapters/full fine-tuning, it is a local NTK approximation. The same expression reveals that forgetting concentrates in a small number of old-task NTK eigenmodes and under frozen linear heads gives a Kronecker scaling rule for the vulnerable rank. These results clarify the relation to prior NTK-overlap theory, explain why parameter-space regularizers can miss output-space interference, and motivate a targeted spectral regularizer.

Inspired by the function-space theory of neural networks, we formulate and analyze a variation space for nonlinear operators between Hilbert spaces, defined through vector-valued Borel measures of bounded variation. We characterize its unit ball as the closed convex hull of a vector-valued single-neuron dictionary in Bochner spaces. For the ReLU activation, the bounded linear operators in this space are precisely the Schatten-$1$ operators, with equivalent norms. For operators in this space, we establish encoder--decoder approximation bounds in the Bochner $L^q$-norm, where the error decomposes into input and output encoding errors and a finite-width term of order $N^{-1/2}$. Under sub-Gaussian assumptions on the input and noise, we further derive high-probability generalization bounds for empirical least squares over path-norm-constrained encoder--decoder networks; the finite-sample contribution to the squared prediction error is of order $K^{-1/2}$ up to logarithmic factors. The finite-width and finite-sample constants are independent of the encoding dimensions and bases, with the latter also independent of the network width. When the encoding errors decay algebraically, these bounds yield algebraic approximation and learning rates, in contrast to the complexity barriers for Lipschitz and Fréchet differentiable operator classes.

Many machine learning techniques rely on approximating a loss function's curvature, but this is notoriously hard to do at the scale of modern deep networks. Surprisingly, no previous work has exploited the curvature constraints that arise from well known weight-space symmetries in loss landscapes. By analytically averaging over group actions that leave the loss invariant, we construct structured Hessian approximations from single gradients that can be tractably estimated, stored, and inverted. The choice of user-specified symmetry group directly governs the trade-off between approximation accuracy and computational cost. Moreover, our framework provides a unifying theoretical lens for viewing existing methods; in particular, a specific choice of symmetry group recovers Shampoo/Muon-like curvature estimates. We validate our method on a range of network architectures, and deploy it to second-order optimization benchmarks, including a small language model. Our curvature estimation framework might find applications in other machine learning problems such as uncertainty estimation, continual learning, compression/pruning, training data attribution, and more.

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.

Mechanistic interpretability aims to break models into meaningful parts; verifying that two such parts implement the same computation is a prerequisite. Existing similarity measures evaluate either empirical behaviour, leaving them blind to out-of-distribution mechanisms, or basis-dependent parameters, meaning they disregard weight-space symmetries. To address these issues for the class of tensor-based models, we introduce a weight-based metric, tensor similarity, that is invariant to such symmetries. This metric captures global functional equivalence and accounts for cross-layer mechanisms using an efficient recursive algorithm. Empirically, tensor similarity tracks functional training dynamics, such as grokking and backdoor insertion, with higher fidelity than existing metrics. This reduces measuring similarity and verifying faithfulness into a solved algebraic problem rather than one of empirical approximation.

Recent advances in neural operators have made partial differential equation (PDE) surrogate modeling increasingly scalable and transferable through large-scale pretraining and in-context adaptation. However, after a shared operator is fine-tuned to multiple regimes within a continuous physical family, it remains unclear whether the resulting weight-space updates merely form isolated regime experts or reveal reusable physical structure. Starting from a shared family anchor, we fine-tune low- and high-regime endpoint experts and show that their updates can be separated into a family-shared adaptation and a direction aligned with the underlying physical parameter. This separation reinterprets endpoint experts as finite-difference probes of a local physical direction in weight space, explaining why static averaging can interpolate between regimes but attenuates endpoint-specific physics. Building on this perspective, we propose Calibration-Conditioned Merge (CCM), a post-hoc coordinate readout method for composing neural PDE experts along this physical direction. Given physical metadata, a calibrated coordinate mapping, or a short observed rollout prefix, CCM infers the target composition coordinate and deploys a single merged checkpoint for the remaining rollout. We evaluate CCM on the reaction--diffusion system, viscosity-parameterized two-dimensional Navier--Stokes equations, and radial dam-break dynamics. Across these benchmarks, CCM achieves its strongest gains in extrapolative regimes, reducing out-of-distribution rollout error relative to the family anchor by 54.2%, 42.8%, and 13.8%, respectively. Further experiments across FNO scales, a DPOT-style backbone, and ablations confirm that endpoint fine-tuning is not arbitrary checkpoint drift, but reveals a calibratable physical direction for training-free transfer across PDE regimes.

Modern agents built on frontier language models often cannot adapt their weights. What, then, remains trainable? We argue it is the agent's \emph{workspace}, the structured external substrate it reads, writes, and tests; we call its evolution workspace optimization. Workspace optimization targets hard multi-turn environments where a frontier model has strong priors but cannot solve the task in a single shot, so the agent must learn through interaction. We propose a principled way to evolve the workspace, mirroring the structure of weight-space training: artifacts in place of parameters, evidence in place of data, counterexamples in place of losses, and textual feedback in place of gradients. We instantiate the idea in DreamTeam, a multi-agent harness for ARC-AGI-3 whose roles build an executable world model, plan, hypothesize, probe, strategize, and route failures. On the current 25-game ARC-AGI-3 public set under the official scoring protocol and averaged over two independent runs, DreamTeam improves the SOTA protocol-matched agent's score from 36% to 38.4%, while using 31% fewer environment actions per game.

Low-Rank Adaptation (LoRA) enables parameter-efficient fine-tuning of large models by decomposing weight updates into low-rank matrices, significantly reducing storage and computational overhead. While effective, standard LoRA lacks mechanisms for uncertainty quantification, leading to overconfident and poorly calibrated models. Bayesian variants of LoRA address this limitation, but at the cost of a significantly increased number of trainable parameters, partially offsetting the original efficiency gains. Additionally, these models are harder to train and may suffer from unstable convergence. In this work, we propose a novel framework for parameter-efficient Bayesian fine-tuning, demonstrating that effective uncertainty quantification can be achieved in very low-dimensional parameter spaces. The proposed method achieves strong performance with improved calibration and generalization while maintaining computational efficiency. Our empirical findings show that, with the appropriate projection of the weight space uncertainty can be effectively modeled in a low-dimensional space, and weight covariances exhibit low ranks.

Overparameterization is central to the success of deep learning, yet the mechanisms by which it improves optimization remain incompletely understood. We analyze weight-space symmetries in neural networks and show that overparameterization introduces additional symmetries that benefit optimization in two distinct ways. First, we prove that these symmetries act as a form of diagonal preconditioning on the Hessian, enabling the existence of better-conditioned minima within each equivalence class of functionally identical solutions. Second, we show that overparameterization increases the probability mass of global minima near typical initializations, making these favourable solutions more reachable. These results offer a potential link between loss landscape geometry and simplicity bias. Empirically, we observe wider networks have lower top eigenvalues, smaller condition numbers and faster convergence, matching our analysis. Our analysis provides a unified framework for understanding overparameterization and width growth as a geometric transformation of the loss landscape.

Weight-space model merging combines independently fine-tuned checkpoints without access to the original training data. While merging has shown promise in multitask settings, its behavior in multilingual generative systems remains underexplored. We systematically study weight-space merging for multilingual machine translation by fully fine-tuning language models on large-scale bilingual corpora and evaluating representative merging strategies across shared-source, shared-target, and bidirectional consolidation settings. Our experiments reveal a strong directional asymmetry. Merging is comparatively more effective when models share a target language, improving multilingual coverage over the base model, but it still fails to preserve the peak performance of language-specific checkpoints. In contrast, when target languages differ, performance degrades sharply, especially in shared-source and bidirectional settings. To explain this behavior, we analyze internal representations and find that fine-tuning does not create disjoint language-specific sub-networks. Instead, independently fine-tuned models activate largely overlapping neurons while reshaping upper-layer target-generation representations into incompatible geometries. These findings suggest that multilingual merging failures arise from target-side geometric misalignment within shared computational units, challenging the assumptions underlying standard weight-space merging for multilingual translation. We make the code publicly available at https://github.com/babangain/mt-model-merging

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.

Autoencoders have found widespread application in both their original deterministic form and in their variational formulation (VAEs). In scientific applications and in image processing it is often of interest to consider data that are viewed as functions; while discretisation (of differential equations arising in the sciences) or pixellation (of images) renders problems finite dimensional in practice, conceiving first of algorithms that operate on functions, and only then discretising or pixellating, leads to better algorithms that smoothly operate between resolutions. In this paper function-space versions of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced, analysed, and deployed. Well-definedness of the objective governing VAEs is a subtle issue, particularly in function space, limiting applicability. For the FVAE objective to be well defined requires compatibility of the data distribution with the chosen generative model; this can be achieved, for example, when the data arise from a stochastic differential equation, but is generally restrictive. The FAE objective, on the other hand, is well defined in many situations where FVAE fails to be. Pairing the FVAE and FAE objectives with neural operator architectures that can be evaluated on any mesh enables new applications of autoencoders to inpainting, superresolution, and generative modelling of scientific data.