SILVA Networks as Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields

Deep networks are powerful function approximators, but they typically store many different computations in shared weight matrices, making it difficult to selectively reuse or adapt parts of them when a familiar structure appears in novel combinations. We introduce the Vector Network (VN), a hierarchical recurrent architecture in which each layer replaces a fixed weight matrix with a library of reusable rank-1 weight atoms. For each input, VN minimizes a layer-local energy to infer a sparse set of active weight atoms and their coefficients, jointly constrained by bottom-up input reconstruction and top-down feedback consistency. These weight atom coefficients then compose an input-specific low-rank weight matrix for that sample. After convergence, slow learning updates only the selected weight atoms through local residual signals scaled by the inferred coefficients. We evaluate VN on four compositional benchmarks spanning 1D signals, 2D spatial decoding, N-body dynamics, and compositional MNIST. VN matches strong baselines in distribution while often achieving out-of-distribution error about an order of magnitude lower when familiar factors must be recombined in novel ways. Vector networks thus make compositional generalization a structural property of the architecture and inference process rather than a brittle byproduct of fitting many behaviors into one shared dense parameter substrate.

Transfer in coordinate networks is often measured by warm-start gain, but whether that gain reflects source-specific structure or generic weight reuse is less clear. We study this question across three implicit neural representation (INR) families, SIREN, ReLU MLPs, and Fourier-feature MLPs, using controlled analytic tests, a 2D lid-driven-cavity Navier--Stokes benchmark, and 1D PDE reference-solution suites for heat, viscous Burgers, and focusing cubic NLS. The analytic tests use independent-seed random controls, while the PDE benchmarks use alternate same-family source controls and auxiliary ablations. Across settings, transfer magnitude and transfer specificity separate clearly. In a 10-seed controlled 1D geometric test, Fourier Features show the largest structured transfer ($33.1\times$), followed by SIREN ($23.0\times$) and ReLU ($10.7\times$), but ReLU is far more selective: random-control transfer is $0.41\times$ for ReLU versus $14.24\times$ for SIREN. On a controlled two-parameter 1D family, the ranking changes: ReLU gives the clearest structured-versus-control separation at default settings, whereas Fourier Features improve only after bandwidth retuning. In Navier--Stokes and the broader 1D PDE suite, no single architecture dominates every equation, yet the same pattern remains: SIREN often reuses weights broadly, whereas ReLU and, in some equations, Fourier Features are more source-selective. Static diagnostics remain weak, and the heuristic scaling law $A_{\text{transfer}} \propto 1/Δt^2$ is rejected in the implemented 1D audit. These results position transfer specificity as a useful diagnostic for coordinate networks and suggest that architecture selection in scientific machine learning should be evaluated under explicit control conditions, not by transfer magnitude alone.

We investigate a structure-first approach to dynamical learning in which the organization of stateful interactions is prescribed explicitly rather than left entirely to a generic recurrent parameterization. We introduce causal recurrent units built from an ordered sequence of local, state-modulated transformations. The construction is motivated by wave-based interaction models, but the units studied here do not impose scattering, passivity, or energy-balance constraints. Because fixed recurrent dynamics, designed reservoir topologies, readout-only learning, and recurrent depth are already well established, the empirical question is deliberately narrower: does the proposed interaction organization provide a useful inductive bias under controlled computational conditions? We compare a one-layer structured model, a two-layer structured model, and a generic echo-state network (ESN), all with 12 recurrent states and the same strictly linear ridge readout. Each model family receives the same random-search budget on calibration data that are disjoint from the final test data, after which the selected hyperparameters are frozen. On a custom nonlinear identification task, the one-layer structured model attains a mean validation NMSE of 2.76 x 10^{-4}, compared with 3.19 x 10^{-4} for the two-layer model and 3.94 x 10^{-4} for the ESN. On NARMA10 the ordering reverses: the ESN attains 0.312, compared with 0.348 and 0.357 for the one- and two-layer structured models. Thus, the proposed organization can be competitive and advantageous on one task, but it is not universally superior; moreover, recurrent depth does not provide a systematic benefit under matched state dimension. The results support a task-dependent interpretation of structural inductive bias and position the present architecture as a controlled precursor to stronger wave- and system-theoretic constructions.