Solves: Scalability

Machine-learning ideas tagged Scalability in the Solves taxonomy of the Math2NN corpus.

Unverified 2026

Morphogen-Controlled Topology-Preserving Sparse Growth

Replace fixed sparse masks with a stochastic birth-death process for neural connections or spatial units. A diffusing morphogen-like utility field controls where connections are added or removed, while a local simple-point test rejects removals or additions that would disconnect a layer or alter a prescribed computational topology. This creates an adaptive sparse architecture with a tunable compact-to-branched transition rather than unconstrained magnitude pruning.

Useful6/10
Difficulty6/10
Novelty6/10
Paper: Control of morphology and topology in a lattice model of branching morphogenesis arXiv:2607.24619
Unverified 2026

Decoder-Side Refinement for Heavy-Tailed Gradients

Replace ordinary sign-gradient or fixed-range gradient quantization in distributed training with a non-adaptive one-bit refinement block. Workers generate all localization and multiscale residue bits using public randomness before the server knows the current aggregate gradient; after decoding a coarse center, the server chooses which stored phase and scale interpretation to use. This should reduce clipping and variance for heavy-tailed gradients without requiring a second communication round.

Useful6/10
Difficulty6/10
Novelty7/10
Paper: Universal Refinement without Interaction: Order-Optimal 1-Bit Mean Estimation arXiv:2607.24358
Unverified 2026

Matching-safe sparse layers

Construct the connectivity mask of a sparse MLP or MoE layer so that every active feature group is covered by a matching to an independent input or sample group. If the mask contains unmatched vertices, repair it with the fewest additional edges or low-rank skip connections before training. The goal is to avoid width- or sparsity-induced singular regimes that can produce sharp interpolation-like loss and generalization spikes.

Useful6/10
Difficulty5/10
Novelty8/10
Paper: The Zero Pattern of a Design Matrix Drives Multiple Descent in Over-parameterized Regression arXiv:2607.24041
Unverified 2026

Effective-Rank Spectral Bottleneck

Use the paper's effective-rank and eigengap-dependent covariance estimation rates to construct a confidence-aware low-rank bottleneck for transformer activations or key/value tensors. The bottleneck is enabled only when the top-p empirical eigenspace can be estimated more accurately than the desired compression error; otherwise the layer remains full-rank.

Useful6/10
Difficulty5/10
Novelty5/10
Paper: Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency arXiv:2607.23964
Unverified 2026

Physical-Adjoint Dual-Weighted PINN

Train a primal neural PDE solver and a separate physical-adjoint neural solver, then use their first-order-system residuals to adaptively allocate collocation points toward regions that control a chosen quantity of interest. Instead of minimizing only the primal residual uniformly, prioritize points according to a balanced combination of primal and adjoint local residuals, because the target-output error is controlled by their global product.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: Goal-Oriented Error Estimation for Least-Squares Finite Element Methods via Physically Meaningful Adjoint PDEs arXiv:2607.23850
Unverified 2026

Mean-Payoff Tropical Router

Replace the usual independently normalized MoE router scores with a small system of tropical inequalities controlling expert load, score margins, and capacity slack. Each inequality induces a local max-plus policy selecting its currently dominant expert or constraint; policy improvement detects positive-weight cycles that would cause oscillatory routing and applies the smallest bias correction that removes them. This provides a non-differentiable but cheap controller around the router…

Useful6/10
Difficulty6/10
Novelty6/10
Paper: Set-defined graph classes: $χ$-boundedness meets tropical algebra arXiv:2607.23754
Unverified 2026

Regular Hypergraph Masks for Balanced Sparse Attention

Construct sparse attention masks as regular or almost-regular hypergraphs: every query selects exactly w keys, while every key receives nearly equal incidence within each score or locality class. This preserves a fixed sparsity budget while removing systematic positional and head-level load imbalance that can cause hot keys, uneven memory traffic, and unstable gradients.

Useful6/10
Difficulty6/10
Novelty6/10
Paper: Optimal Dynamic Cooling of Multiple Qubits arXiv:2607.23641
Unverified 2026

Connectivity-aware ATC gradient tracking

Use the paper's mode decomposition to choose between ordinary DIGing and augmented ATC-DIGing/AugDGM according to the communication spectrum and curvature. The augmented scheme changes the disagreement feedback gain mode by mode, which can reduce the dominant pole radius on well-connected graphs without changing the neural-network architecture.

Useful6/10
Difficulty5/10
Novelty5/10
Paper: Exact Worst-case Convergence Rates of Distributed Gradient Tracking Methods arXiv:2607.23601
Unverified 2026

Quadratic-anisotropy electrostatic surrogate

Use the small dielectric-anisotropy estimate to replace expensive nonlocal electrostatic solves with a local field-energy surrogate during most neural-network updates. Periodically evaluate the exact field, estimate the approximation constant, and trigger correction solves only when the observed error exceeds the predicted quadratic scale.

Useful6/10
Difficulty4/10
Novelty7/10
Paper: Variational principles for the interaction of liquid crystals and electric fields in the Oseen--Frank model arXiv:2607.23315
Unverified 2026

Homomorphic Tempo Conditioning

Condition a temporal neural network on a tempo or dilation ratio through a homomorphism from multiplicative positive scales to additive latent shifts. A ratio composed from several scale changes then produces the sum of their learned effects, allowing interpolation and extrapolation to rates absent from training instead of using an independent embedding per rate.

Useful6/10
Difficulty4/10
Novelty5/10
Paper: Infinite Canons: Maximally Self-Similar Melodic Lines and Canons with Infinite Solutions arXiv:2607.23210
Unverified 2026

Augmented KKT preconditioner for constrained neural training

Replace a generic optimizer for equality-constrained neural-network updates with an augmented KKT solve that separately preconditions parameter and constraint variables. Add a positive semidefinite constraint penalty to the parameter block; this does not change an exactly feasible solution but makes the primal block better conditioned and the constraint Schur complement more robust.

Useful6/10
Difficulty6/10
Novelty6/10
Paper: Optimal block preconditioners for a mass-conserving mixed stress formulation of Stokes flow arXiv:2607.22932
Unverified 2026

Log-Rank Fluctuation Channels

Augment a causal sequence model with a small hierarchy of prefix summaries weighted by powers of the logarithmic rank of each preceding token. The summaries retain order-sensitive deviations from a baseline representation while costing O(KNd) for sequence length N, hierarchy width K, and hidden dimension d, instead of O(N^2d) dense attention.

Useful6/10
Difficulty4/10
Novelty6/10
Paper: A Logarithmic Fluctuation Hierarchy for Sequential Interacting Diffusions arXiv:2607.22470
Unverified 2026

Berrut-Coded Neural Evaluation

Encode K input vectors at Chebyshev nodes with a Berrut rational interpolant, send different evaluations of that interpolant to N workers, and let every worker run the same neural function on its encoded input. The master reconstructs the desired outputs at the original data nodes from worker responses, allowing stragglers to be omitted and enabling robust rejection of inconsistent responses.

Useful6/10
Difficulty6/10
Novelty7/10
Paper: Robust Berrut-Approximated Coded Computing via Discrete Cosine Transforms arXiv:2607.22427
Unverified 2026

Refine-and-Coarsen Block Router

Turn the paper's graphon refinement operation into an adaptive mixture-of-experts or relational-attention router whose number of blocks is learned rather than fixed. Split a block when its members produce incompatible constraint gradients, and merge blocks when their relation rows are statistically indistinguishable, yielding a compact architecture that grows only where the data require extra structure.

Useful6/10
Difficulty5/10
Novelty7/10
Paper: Constrained Multi-Relational Graphons with Maximum Entropy arXiv:2607.22383
Unverified 2026

Superlinear Reinforced MoE Routing

Replace one-shot softmax expert routing with a small number of synchronous routing rounds in which each token resamples an expert with probability proportional to that expert's current load raised to a power \(\alpha>1\). The resulting positive feedback rapidly creates a dominant routing basin, potentially reducing the number of active experts and communication groups at inference while retaining a controllable exploration phase through the initial round or a token-specific score factor.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: The Conclave Process arXiv:2607.22324
Unverified 2026

Geometric-Consensus Q-Network Training

Train identical Q-networks on multiple agents using local TD statistics and communicate only through periodic multi-step consensus. Within each epoch, agents perform local updates and then apply L mixing rounds to the vector of Q-values, TD targets, or parameter deltas; choose L so that the residual disagreement is below the stochastic estimation error. The method targets communication reduction at fixed sample efficiency, especially when N agents collect experience in parallel.

Useful6/10
Difficulty5/10
Novelty5/10
Paper: Variance-Reduced Q-Learning over Static and Time-Varying Networks arXiv:2607.21876
Unverified 2026

Marchenko–Pastur Tensor Initialization

Use Marchenko–Pastur spectral edges to calibrate tensorized random features even when the base vector has exchangeable, sign-symmetric dependent coordinates. Rescale the tensor features and select their retained dimension so the predicted covariance bulk remains well-conditioned instead of assuming independent Gaussian coordinates.

Useful6/10
Difficulty5/10
Novelty7/10
Paper: Marchenko-Pastur law for tensor powers of exchangeable unconditional vectors arXiv:2607.21759
Unverified 2026

Onsager-Coupled Block Optimizer

Use a symmetric positive-definite, non-diagonal mobility matrix to couple updates of parameter groups, analogous to drag-modified Onsager mobility coupling ionic species. Estimate local block curvature and select the learning rate from the generalized spectrum of mobility times curvature, targeting rapid loss decay without the instability of aggressively scaled diagonal optimizers.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: Global Finite-Energy Weak Solutions and Sharp Entropy Decay for a Poisson-Nernst-Planck System with Interspecies Drag and Steric Effects arXiv:2607.21742
Unverified 2026

Subcritical Blob Graph Sparsifier

Turn stochastic edge thinning into a mass-aware graph sparsifier: first form retained connected blobs, then connect blob summaries using the rank-one shortcut probability from the paper. Choose the retention rate below the explicit subcritical threshold so the contracted graph remains a collection of small decorations around leading blobs, reducing message-passing cost without creating an artificial giant component.

Useful6/10
Difficulty6/10
Novelty6/10
Paper: Subcritical percolation and network archaeology on random recursive tree substrate networks arXiv:2607.21428
Unverified 2026

Approximation-Aware Hard-Core Routing

Construct a sparse routing or graph-neural architecture whose activation gates satisfy a hard-core constraint: neighboring sites, experts, or token groups cannot be active simultaneously. Compare the same local routing rule on bipartite and random regular interaction graphs; the graph structure should change the maximum usable activation dimension and may also change optimization stability.

Useful6/10
Difficulty5/10
Novelty7/10
Paper: A Hard-Core Subshift Whose Sofic Mean Dimension Depends on the Sofic Approximation arXiv:2607.21398
Unverified 2026

Free-volume load balancing for sparse experts

Model each expert as a cell with occupancy q_i, vacancy n_i = 1 − q_i, and a nonzero localization floor Δ_i. Add a free-volume potential whose derivative becomes strong when an expert is poorly utilized, but remains finite because of Δ_i. Unlike ordinary entropy balancing, this mechanism predicts a quantitative inverse-vacancy regime and a measurable crossover to saturation.

Useful6/10
Difficulty4/10
Novelty5/10
Paper: Free-volume origin of diverging direct correlations in hard crystals: insights from an exact one-dimensional model arXiv:2607.21379
Unverified 2026

Entropy-Calibrated Non-Backtracking Message Passing

Replace ordinary graph propagation, which repeatedly revisits the edge it just traversed, with a directed-edge non-backtracking operator. Normalize its learned gain using an estimate of the Hashimoto spectral radius so that feature magnitudes neither explode on high-growth graphs nor vanish on sparse graphs.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: Critical-exponent spectra and rank two inverse realization on biregular trees arXiv:2607.21294
Unverified 2026

Multiplicative Adaptive Attention Graph

Give each query-token pair a positive adaptive edge weight that evolves by a multiplicative rule instead of relying only on instantaneous dot-product attention logits. Edges whose aggregate interaction is useful can grow, while overloaded or incompatible neighborhoods can shrink. Sparse initialization is preserved because an edge initialized at zero remains zero under the multiplicative dynamics.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: The mean-field limit of non-exchangeable particle systems with non-conservative dynamics and adaptive weights arXiv:2607.21110
Unverified 2026

Reflected Survival Routing

Replace independent binary early-exit or token-pruning decisions with a monotone randomized survival process for each token or expert route. A token can lose survival mass at each layer but cannot become active again; the model is trained with a reflected obstacle-style penalty that activates when the predicted value of continuing computation is below the value of stopping plus the compute cost. Mean-field statistics are computed over currently surviving tokens, making routing less sensitive to…

Useful6/10
Difficulty5/10
Novelty5/10
Paper: A new probabilistic approach for mean field games of optimal stopping arXiv:2607.21062