Feed-Forward Network

This paper investigates the learning theory of Transformer networks for regression tasks on the compact Euclidean domain $[0,1]^d$ and $d$-dimensional compact Riemannian manifolds. We propose a novel constructive approximation framework for Transformers that builds local approximations of the target function and aggregates them into a global approximation via softmax partition of unity. This approach leverages the attention mechanism to achieve spatial localization through affine transformations of the input. The softmax activation plays a crucial role in aggregating local approximations to a global output. From an approximation perspective, we prove that a dense Transformer equipped with only two encoder blocks and standard single-hidden-layer point-wise feed-forward networks can achieve a uniform $\varepsilon$-approximation error for $α$-Hölder continuous functions with $α\in (0,1]$ using $\mathcal{O}(\varepsilon^{-d/α})$ total parameters. Building upon this approximation guarantee, we establish a near minimax-optimal generalization error bound of order $\mathcal{O}\big(n^{-\frac{2α}{2α+d}} \log n\big)$ for the empirical risk minimizer, where $n$ is the training data size. The Transformer architecture studied in this paper is dense, shallow and wide, and employs softmax activation and sinusoidal positional encodings, closely reflecting practical implementations.

As Large Language Models (LLMs) transition into conversational agents, generative advertising emerges as a crucial monetization strategy. However, embedding advertisements within unstructured LLM outputs introduces a critical trilemma: balancing advertiser payoffs, platform revenue, and user experience. Existing methods, such as prompt injection or rigid position slots, disrupt semantic coherence and lack a parametric framework for independent control, rendering rigorous mechanism design intractable. To bridge this gap, we introduce Neuron Auctions, a novel paradigm that shifts the auction object from the surface text space to the LLM's internal representations. Leveraging mechanistic interpretability, we identify brand-specific feed-forward network (FFN) neurons and demonstrate that competing brands activate within approximately orthogonal subspaces. This near-perfect independence allows us to define continuous, disentangled intervention budgets (specifically, neuron counts and amplification factors) as auctionable commodities. Building on this computational carrier, we design a continuous menu-based auction mechanism that naturally guarantees strategy-proofness and optimizes revenue for the platform. By explicitly incorporating a user utility penalty into the platform's optimization objective, our framework dynamically prices out overly aggressive interventions. Extensive experiments demonstrate that Neuron Auctions effectively preserve natural discourse quality while achieving an optimal alignment between commercial incentives and user satisfaction.

We show that, in a precise sense, a broad class of feedforward neural networks learn (have finite sample complexity) in the PAC model: every fixed finite feedforward architecture whose layers are definable in an o-minimal structure has finite sample complexity in the agnostic PAC setting, even with unbounded parameters. This covers standard fixed-size MLPs, CNNs, GNNs, and transformers with fixed sequence length, together with the operations and layers typically used in such architectures, including linear projections, residual connections, attention mechanisms, pooling layers, normalization layers, and admissible positional encodings. Hence, distribution-free learnability for modern non-recurrent architectures is not an exceptional property of particular activations or architecture-specific VC arguments, but a consequence of tame feedforward computation. Our results reposition finite-sample PAC learnability as a baseline rather than a differentiator: they shift the focus of architectural comparison toward inductive biases, symmetries and geometric priors, scalability, and optimization behaviour.

Despite the prevalence of the attention sink phenomenon in Large Language Models (LLMs), where initial tokens disproportionately monopolize attention scores, its structural origins remain elusive. This work provides a \textit{mechanistic explanation} for this phenomenon. First, we trace its root to the value aggregation process inherent in self-attention, which induces a systematic variance discrepancy. We further demonstrate that this discrepancy is drastically amplified by the activation of super neurons within Feed-Forward Network (FFN) layers. Specifically, the channel-sparse down-projections trigger a dimension disparity of the first-token representation, necessitating the formation of attention sinks as a structural anchor. Then, we validate this causal chain through two controlled interventions: (i) isolating the aggregation effect via attention mask modifications and (ii) amplifying the variance of targeted token representations. Both interventions can replicate attention sinks at arbitrary positions. Our mechanistic understanding offers a foundation for the systematic control of sink formation. Finally, as a proof of concept, we propose \textit{head-wise RMSNorm}, an architectural modification that stabilizes value aggregation outputs during pre-training. Our experiments demonstrate that restoring statistical parity across positions significantly accelerates convergence.

Neural networks transform data through learned representations whose geometry affects separation, contraction, and generalization. Recent work studies this geometry using discrete curvature on neighborhood graphs, suggesting Ricci-flow-like behavior across layers. We develop a smooth operator-theoretic alternative for feedforward representation snapshots. Each feature cloud induces a Gaussian-kernel diffusion Markov operator, and transport, spectral, label-boundary, and local-scale observables are derived from this single object via Bakry-Emery $Γ$-calculus. In a balanced Gaussian class-conditional snapshot model with shared covariance, the population operator has closed-form class affinities, leakage, and coarse spectra, all controlled by pairwise regularized Mahalanobis separations $c_\varepsilon^{(a,b)}$. We also prove that the resulting operator observables vary smoothly under feature perturbations, while hard neighborhood-graph diagnostics can change discontinuously. Synthetic experiments validate the closed-form Gaussian bridge, while learned MNIST experiments show that the same operator observables track training, width, and perturbation stability. Together, these results give a stable operator-geometric framework for analyzing feedforward representation geometry.

Perturbation probing generates task-specific causal hypotheses for FFN neurons in large language models using two forward passes per prompt and no backpropagation, followed by a one-time intervention sweep of about 150 passes amortized across all identified neurons. Across eight behavioral circuits, 13 models, and four architecture families, we identify two circuit structures that organize LLM behavior. Opposition circuits appear when RLHF suppresses a pre-training tendency. In safety refusal, about 50 neurons, or 0.014 percent of all neurons, control the refusal template; ablating them changes 80 percent of response formats on 520 AdvBench prompts while producing near-zero harmful compliance, 3 of 520 cases, all with disclaimers. Routing circuits appear for pre-training behaviors distributed through attention. For language selection, residual-stream direction injection switches English to Chinese output on 99.1 percent of 580 benchmark prompts in the 3 of 19 tested models that satisfy three observed conditions: bilingual training, FFN-to-skip signal ratio between 0.3 and 1.1, and linear representability. The same intervention fails on the other 16 models and on math, code, and factual circuits, defining the limits of directional steering. The FFN-to-skip signal ratio, computed from the same two forward passes, distinguishes the two structures and predicts the appropriate intervention. Circuit topology varies by architecture, from Qwen's concentrated FFN bottleneck to Gemma's normalization-shielded circuit. In Qwen3.5-2B, ablating 20 neurons eliminates multi-turn sycophantic capitulation, while amplifying 10 related neurons improves factual correction from 52 percent to 88 percent on 200 TruthfulQA prompts. These results show that perturbation probing offers mechanistic insight into RLHF-organized behavior and a practical toolkit for precision template-layer editing.

I propose the \emph{Random Cloud} method, a training-free approach to neural architecture search that discovers minimal feedforward network topologies through stochastic exploration and progressive structural reduction. Unlike post-training pruning methods that require a full train-prune-retrain cycle, this method evaluates randomly initialized networks without backpropagation, progressively reduces their topology, and only trains the best minimal candidate at the end. I evaluate on 7 classification benchmarks against magnitude pruning and random pruning baselines. The Random Cloud matches or outperforms both baselines in 6 of 7 datasets, achieving statistically significant improvements on Sonar ($+4.9$pp accuracy, $p{=}0.017$ vs magnitude pruning) with 87% parameter reduction. Crucially, the method is faster than both pruning baselines in 4 of 5 datasets (0.67--0.94$\times$ the cost of full training), since it avoids training the full-size network entirely.

These lecture notes provide an introduction to the verification of neural networks from a theoretical perspective. We discuss feed-forward neural networks, recurrent neural networks, attention mechanisms, and transformers, together with specification languages and algorithmic verification techniques.

We investigate whether the Feed-Forward Network (FFN) sublayer in a decoder-only transformer can be replaced by an explicit learned memory graph while preserving the surrounding autoregressive architecture. The proposed Graph Memory Transformer (GMT) keeps causal self-attention intact, but replaces the usual per-token FFN transformation with a memory cell that routes token representations over a learned bank of centroids connected by a learned directed transition matrix. In the base GMT v7 instantiation studied here, each of 16 transformer blocks contains 128 centroids, a 128 * 128 edge matrix, gravitational source routing, token-conditioned target selection, and a gated displacement readout. The cell therefore returns movement from an estimated source memory state toward a target memory state, rather than a retrieved value. The resulting model is a fully decoder-only language model with 82.2M trainable parameters and no dense FFN sublayers, compared with a 103.0M-parameter dense GPT-style baseline used in the evaluation. The base v7 model trains stably and exposes centroid usage, transition structure, and source-to-target movement as directly inspectable quantities of the forward computation. It remains behind the larger dense baseline in validation loss and perplexity (3.5995/36.58 vs. 3.2903/26.85), while showing close zero-shot benchmark behavior under the evaluated setting. These results are not intended as a state-of-the-art claim; they support the viability and structural interpretability of replacing dense within-token transformation with graph-mediated memory navigation. Broader scaling, optimized kernels, and more extensive benchmark evaluation are left for subsequent work.

A widely cited result by Dong et al. (2021) showed that Transformers built from self-attention alone, without skip connections or feed-forward layers, suffer from rapid rank collapse: all token representations converge to a single direction. The proposed remedy was the MLP. We show that this picture, while correct in the regime studied by Dong, is incomplete in ways that matter for architectural understanding. Three results are established. First, layer normalisation is precisely affine-rank-neutral: it preserves the affine rank of the token representation set exactly. The widespread claim that LN "plays no role" is imprecise; the correct statement is sharper. Second, residual connections generically obstruct rank collapse in real Transformers such as BERT-base, in a measure-theoretic sense, without contribution from the MLP. The MLP's irreplaceable function is different: generating feature directions outside the linear span of the original token embeddings, which no stack of attention layers can produce. Third, a phenomenon distinct from rank collapse is identified: head-channel non-identifiability. After multi-head attention sums per-head outputs through the output projection, individual contributions cannot be canonically attributed to a specific head; n(H-1)d_k degrees of freedom per layer remain ambiguous when recovering a single head from the mixed signal. The MLP cannot remedy this because it acts on the post-summation signal. A constructive partial remedy is proposed: a position-gated output projection (PG-OP) at parameter overhead below 1.6% of the standard output projection. The four collapse phenomena identified in the literature -- rank collapse in depth, in width, head-channel non-identifiability, and entropy collapse -- are unified under a symmetry-breaking framework, each corresponding to a distinct symmetry of the Transformer's forward pass.

We study the organization of channel-level importance in transformer feed-forward networks (FFNs). Using a Fisher-style loss proxy (LP) based on activation-gradient second moments, we show that loss sensitivity is concentrated in a small set of channels within each layer. In Llama-3.1-8B, the top 1% of channels per layer accounts for a median of 58.7% of LP mass, with a range of 33.0% to 86.1%. We call these loss-critical channels supernodes. Although FFN layers also contain strong activation outliers, LP-defined supernodes overlap only weakly with activation-defined outliers and are not explained by activation power or weight norms alone. Around this core, we find a weaker but consistent halo structure: some non-supernode channels share the supernodes' write support and show stronger redundancy with the protected core. We use one-shot structured FFN pruning as a diagnostic test of this organization. At 50% FFN sparsity, baselines that prune many supernodes degrade sharply, whereas our SCAR variants explicitly protect the supernode core; the strongest variant, SCAR-Prot, reaches perplexity 54.8 compared with 989.2 for Wanda-channel. The LP-concentration pattern appears across Mistral-7B, Llama-2-7B, and Qwen2-7B, remains visible in targeted Llama-3.1-70B experiments, and increases during OLMo-2-7B pretraining. These results suggest that LLM FFNs develop a small learned core of loss-critical channels, and that preserving this core is important for reliable structured pruning.

The advent of Large Language Models (LLMs) has fundamentally reshaped the way we interact with graphs, giving rise to a new paradigm called GraphLLM. As revealed in recent studies, graph learning can benefit from LLMs. However, we observe limited benefits when we directly utilize LLMs to make predictions for graph-related tasks within GraphLLM paradigm, which even yields suboptimal results compared to conventional GNN-based approaches. Through in-depth analysis, we find this failure can be attributed to LLMs' limited capability for processing graph data and their tendency to overlook graph information. To address this issue, we propose LoReC (Look, Remember, and Contrast), a novel plug-and-play method for GraphLLM paradigm, which enhances LLM's understanding of graph data through three stages: (1) Look: redistributing attention to graph; (2) Remember: re-injecting graph information into the Feed-Forward Network (FFN); (3) Contrast: rectifying the vanilla logits produced in the decoding process. Extensive experiments demonstrate that LoReC brings notable improvements over current GraphLLM methods and outperforms GNN-based approaches across diverse datasets. The implementation is available at https://github.com/Git-King-Zhan/LoReC.

Transformers encode structure in sequences via an expanding contextual history. However, their purely feedforward architecture fundamentally limits dynamic state tracking. State tracking -- the iterative updating of latent variables reflecting an evolving environment -- involves inherently sequential dependencies that feedforward networks struggle to maintain. Consequently, feedforward models push evolving state representations deeper into their layer stack with each new input step, rendering information inaccessible in shallow layers and ultimately exhausting the model's depth. While this depth limit can be bypassed by dynamic depth models and by explicit or latent thinking that externalizes state representations, these solutions are computationally and memory inefficient. In this article, we argue that temporally extended cognition requires refocusing from explicit thought traces to implicit activation dynamics via recurrent architectures. We introduce a taxonomy of recurrent and continuous-thought transformer architectures, categorizing them by their recurrence axis (depth versus step) and their ratio of input tokens to recurrence steps. Finally, we outline promising research directions, including enhanced state-space models and coarse-grained recurrence, to better integrate state tracking into modern foundation models.

Mixture-of-Experts (MoE) models substantially improve performance by increasing the capacity of dense architectures. However, directly training MoE models requires considerable computational resources and introduces extra overhead in parameter storage and deployment. Therefore, it is critical to develop an approach that leverages the multi-expert capability of MoE to enhance performance while incurring minimal additional cost. To this end, we propose a novel pre-training approach, termed ExFusion, which improves the efficiency of Transformer training through multi-expert fusion. Specifically, during the initialization phase, ExFusion upcycles the feed-forward network (FFN) of the Transformer into a multi-expert configuration, where each expert is assigned a weight for later parameter fusion. During training, these weights allow multiple experts to be fused into a single unified expert equivalent to the original FFN, which is subsequently used for forward computation. As a result, ExFusion introduces multi-expert characteristics into the training process while incurring only marginal computational cost compared to standard dense training. After training, the learned weights are used to integrate multi-experts into a single unified expert, thereby eliminating additional overhead in storage and deployment. Extensive experiments on a variety of computer vision and natural language processing tasks demonstrate the effectiveness of the proposed method.

We present UniQueR, a unified query-based feedforward framework for efficient and accurate 3D reconstruction from unposed images. Existing feedforward models such as DUSt3R, VGGT, and AnySplat typically predict per-pixel point maps or pixel-aligned Gaussians, which remain fundamentally 2.5D and limited to visible surfaces. In contrast, UniQueR formulates reconstruction as a sparse 3D query inference problem. Our model learns a compact set of 3D anchor points that act as explicit geometric queries, enabling the network to infer scene structure, including geometry in occluded regions--in a single forward pass. Each query encodes spatial and appearance priors directly in global 3D space (instead of per-frame camera space) and spawns a set of 3D Gaussians for differentiable rendering. By leveraging unified query interactions across multi-view features and a decoupled cross-attention design, UniQueR achieves strong geometric expressiveness while substantially reducing memory and computational cost. Experiments on Mip-NeRF 360 and VR-NeRF demonstrate that UniQueR surpasses state-of-the-art feedforward methods in both rendering quality and geometric accuracy, using an order of magnitude fewer primitives than dense alternatives.

We present a theory-guided generalized Bayesian methodology for spatio-temporal raster data, which we use to train an ensemble of stochastic feed-forward neural networks with Gaussian-distributed weights. The methodology incorporates the dependence and causal structure of a spatio-temporal Ornstein-Uhlenbeck process into training and inference by enforcing constraints on the design of the data embedding and the related optimization routine. In inference mode, the networks are employed to generate causal ensemble forecasts by applying different initial conditions at different horizons. We call this workflow MMAF-guided learning. Experiments conducted on both synthetic and real data demonstrate that our forecasts remain calibrated across multiple time horizons. Moreover, we show that on such data, shallow feed-forward architectures can achieve performance comparable to, and in some cases better than, convolutional or diffusion deep learning architectures used in probabilistic forecasting tasks.

Vision Transformers require significant computational resources and memory bandwidth, severely limiting their deployment on resource-constraint hardware. Most structured pruning methods reduce theoretical cost effectively, yet they typically operate at a single structural granularity and depend on multi-stage pipelines with importance ranking, auxiliary solvers or post-hoc magnitude thresholding, followed by a separate fine-tuning phase to recover accuracy. We propose Hierarchical Auto-Pruning (HiAP), which casts ViT pruning as a single budget-aware learning problem and jointly allocates sparsity across four granularities in one end-to-end phase. HiAP introduces stochastic Gumbel-Sigmoid gates at macro level (attention heads and FFN blocks) and micro level (intra-head dimensions and FFN neurons), and optimizes them against the task loss together with an analytical MAC cost term. The budget coefficient steers the network to a target compute level while the gates gradually harden into a dense, smaller sub-network at convergence. It does not require importance heuristics, ranking metrics, auxillary solvers or secondary fine-tuning. On ImageNet with DeiT small, HiAP automatically discovers hetergenous architectures, pruning depths, heads, and width by different amount across layers, and reaches competitive accuracies against substantially more complex pruning pipelines at comparable compute from a single training run.

The architectural shape of dense Transformers has remained remarkably stable: narrow-wide-narrow feed-forward networks (FFNs) consume most non-embedding parameters. Motivated by theoretical and empirical evidences that residual wide-narrow-wide (hourglass) MLPs remain expressive despite bottlenecks, we revisit whether this architectural convention is necessary for dense language models. We study Hourglass Transformers, which replace the conventional FFN with residual stacks of hourglass sub-MLPs and use hourglass attention to decouple residual-stream width from attention width. This exposes a practical depth-width trade-off: compressing the FFN intermediate dimension allows wider hidden states and fewer layers at matched parameter budgets. Across model scales from 113M to 8B parameters, Hourglass Transformers achieve language-modeling and downstream performance comparable to conventional Transformers, while improving training compute efficiency by $8.7\%$ at matched average downstream accuracy across the 906M, 3B, and 8B scales. After long-context extension, the 8B Hourglass model also outperforms its matched conventional baseline across 4k-64k context lengths. At 64k context, the reduced attention layer count lowers both computation and KV-cache requirements, yielding up to $1.93\times$ faster token decoding and $50\%$ lower KV-cache memory at the 1B scale. These results identify hourglass structures as a practical architecture-efficiency alternative for compute- and latency-conscious Transformer design.

We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms. A comparative analysis between the numerical/exact solutions of the Burgers' and Eikonal equations, and the same obtained via PINNs is presented. We show that PINN learning provides a different computational pathway compared to standard numerical discretization in approximating essentially the same underlying dynamics of the system. Within this framework, DNNs can be interpreted as discrete dynamical systems whose layer-wise evolution approaches attractors, and multiple parameter configurations may yield comparable solutions, reflecting the degeneracy of the inverse mapping. In contrast to the structured operators associated with finite-difference (FD) procedures, PINNs learn dense parameter representations that are not directly associated with classical discretization stencils. This distributed representation generally involves a larger number of parameters, leading to reduced interpretability and increased computational cost. However, the additional flexibility of such representations may offer advantages in high-dimensional settings where classical grid-based methods become impractical.

The rise of 3D Gaussian Splatting has revolutionized photorealistic 3D asset creation, yet a critical gap remains for their interactive refinement and editing. Existing approaches based on diffusion or optimization are ill-suited for this task, as they are often prohibitively slow, destructive to the original asset's identity, or lack the precision for fine-grained control. To address this, we introduce SplatPainter, a state-aware feedforward model that enables continuous editing of 3D Gaussian assets from user-provided 2D view(s). Our method directly predicts updates to the attributes of a compact, feature-rich Gaussian representation and leverages Test-Time Training to create a state-aware, iterative workflow. The versatility of our approach allows a single architecture to perform diverse tasks, including high-fidelity local detail refinement, local paint-over, and consistent global recoloring, all at interactive speeds, paving the way for fluid and intuitive 3D content authoring.

We challenge the conventional view of neural network pruning as solely a compression technique, demonstrating that one-shot magnitude pruning serves as a powerful implicit regularizer for ASR. Using Whisper-small, we combine gradient- and Fisher-based sensitivity diagnostics with targeted, component-wise pruning. This reveals architectural asymmetries: decoder FFNs are pruning-fragile, whereas decoder self-attention and the last encoder layers contain redundancy that, when removed, improves generalization. Without fine-tuning, pruning 50% of decoder self-attention reduces WER by 2.38% absolute (20.44% relative) on LibriSpeech test-other; pruning the last four encoder layers at 50% instead yields a 1.72% absolute (14.8% relative) improvement. Gains persisted on Common Voice and TED-LIUM datasets. Beyond regularization benefits, our sensitivity-aware approach enables more aggressive one-shot compression. At 40% sparsity, where established global pruning approaches catastrophically fail, our method preserves near-baseline accuracy. This positions pruning as a first-class architectural design tool: knowing where to prune is as important as how much to prune.

Given encoded 3D point cloud geometry available at the decoder, we study the problem of lossy attribute compression in a multi-resolution B-spline projection framework. A target continuous 3D attribute function is first projected onto a sequence of nested subspaces $\mathcal{F}^{(p)}_{l_0} \subseteq \cdots \subseteq \mathcal{F}^{(p)}_{L}$, where $\mathcal{F}^{(p)}_{l}$ is a family of functions spanned by a B-spline basis function of order $p$ at a chosen scale and its integer shifts. The projected low-pass coefficients $F_l^*$ are computed by variable-complexity unrolling of a rate-distortion (RD) optimization algorithm into a feed-forward network, where the rate term is the sparsity-promoting $\ell_1$-norm. Thus, the projection operation is end-to-end differentiable. For a chosen coarse-to-fine predictor, the coefficients are then adjusted to account for the prediction from a lower-resolution to a higher-resolution, which is also optimized in a data-driven manner.

Photorealistic volumetric rendering of CT scans greatly benefits clinical workflows, yet neural approaches such as Neural Radiance Fields (NeRF) and 3D Gaussian Splatting (3DGS) require prohibitive per-scan optimization (hours for NeRF, about 30 minutes for 3DGS), making them impractical in clinical settings. We propose Render-FM, a feedforward model that eliminates this bottleneck by directly regressing 6D Gaussian Splatting (6DGS) parameters from a CT volume in a single 2.8-second forward pass, a 500x speedup over per-scan optimization. To bridge the domain gap between natural scene reconstruction and medical volumetric rendering, we introduce Anatomy-Guided Priming (AGP), which incorporates segmentation masks and transfer functions as structural and appearance priors, information that existing Gaussian splatting methods overlook. Built on an nnU-Net-inspired 3D U-Net trained on diverse CT scans, Render-FM predicts per-voxel 6DGS parameters and supports immediate real-time rendering. Unlike per-scan methods, it generalizes to unseen anatomies, novel transfer functions, and enables compositional organ visualization with zero additional preparation time. Optional 89-second fine-tuning further improves quality, surpassing per-scan optimized baselines. Project page: https://gaozhongpai.github.io/renderfm/.

Large language models (LLMs) integrated with retrieval-augmented generation (RAG) have improved factuality by grounding outputs in external evidence. However, they remain susceptible to unfaithful generation, where outputs contradict retrieved context despite its relevance and accuracy. Existing approaches aiming to improve faithfulness primarily focus on enhancing the utilization of external context, but often overlook the persistent influence of internal parametric knowledge during generation. In this work, we investigate the internal mechanisms behind unfaithful generation and identify a subset of mid-to-deep feed-forward networks (FFNs) that are disproportionately activated in such cases. Building on this insight, we propose Parametric Knowledge Muting through FFN Suppression (ParamMute), a framework that improves contextual faithfulness by suppressing the activation of unfaithfulness-associated FFNs and calibrating the model toward retrieved knowledge. To evaluate our approach, we introduce CoFaithfulQA, a benchmark specifically designed to evaluate faithfulness in scenarios where internal knowledge conflicts with accurate external evidence. Experimental results show that ParamMute significantly enhances faithfulness across both CoFaithfulQA and the established ConFiQA benchmark, achieving substantial reductions in reliance on parametric memory. These findings underscore the importance of mitigating internal knowledge dominance and provide a new direction for improving LLM trustworthiness in RAG. All codes are available at https://github.com/OpenBMB/ParamMute.

We present a feedforward graph architecture in which heterogeneous frozen large language models serve as computational nodes, communicating through a shared continuous latent space via learned linear projections. Building on recent work demonstrating geometric compatibility between independently trained LLM latent spaces~\cite{armstrong2026thinking}, we extend this finding from static two-model steering to end-to-end trainable multi-node graphs, where projection matrices are optimized jointly via backpropagation through residual stream injection hooks. Three small frozen models (Llama-3.2-1B, Qwen2.5-1.5B, Gemma-2-2B) encode the input into a shared latent space whose aggregate signal is injected into two larger frozen models (Phi-3-mini, Mistral-7B), whose representations feed a lightweight cross-attention output node. With only 17.6M trainable parameters against approximately 12B frozen, the architecture achieves 87.3% on ARC-Challenge, 82.8% on OpenBookQA, and 67.2% on MMLU, outperforming the best single constituent model by 11.4, 6.2, and 1.2 percentage points respectively, and outperforming parameter-matched learned classifiers on frozen single models by 9.1, 5.2, and 6.7 points. Gradient flow through multiple frozen model boundaries is empirically verified to be tractable, and the output node develops selective routing behavior across layer-2 nodes without explicit supervision.