Wave Function Backpropagation with Explicit Temporal-Interval Dynamics
Authors: Byunggu Yu, Justin Kim
Organizations: Dept. of Computer Science and Information Technology University of the District of Columbia Washington, DC USA
Abstract
Conventional neural networks learn predominantly through affine transformations followed by nonlinear activations, while elapsed time is often treated as an auxiliary feature or assumed to be uniformly sampled. This paper introduces Wave Function Backpropagation (WFB), a wave-parameterized learning formulation in which neural responses are represented by learnable amplitude, wavenumber, angular frequency, and phase. The formulation associates an observed state with its temporal interval Delta t through the phase of a differentiable spatiotemporal wave. We derive standard WFB gradients and a spatial-curvature correction based on the Laplacian of the wave response. WFB is instantiated in a deliberately feed-forward trajectory predictor to provide a controlled proof of concept; sequence learning is outside the scope of the present evaluation. With motion features, STD-WFB using real intervals reduces average displacement error (ADE) by 20.4% relative to the original FFN baseline. In a new position-only evaluation that removes temporal leakage through precomputed velocity and acceleration, real-interval WFB reduces ADE by 10.4% relative to the original FFN and remains competitive with parameter-matched ReLU controls, obtaining 2.1% lower mean ADE than the matched FFN with explicit Delta t. Shuffled-interval WFB attains the lowest mean ADE, indicating that the present evidence supports the effectiveness of the wave representation but does not attribute the gain to interval alignment. These results establish WFB as a viable structured feed-forward learning formulation and define a clear basis for subsequent architectural studies.
A Noise-modulated Neural Network (NNN) learns and infers only in the presence of noise, treating noise as a computational resource rather than a disturbance. The noise lets it learn efficiently by backpropagation while transmitting spike-like signals, but backpropagation needs a reverse path through transposed weights, the weight transport problem, which undermines biological and neuromorphic plausibility. Forward-only alternatives typically substitute a different objective or fixed random feedback, sacrificing stability and accuracy. We show that backpropagation itself can be reconstructed in the NNN from forward-pass statistics alone: a weight mirror estimates each weight matrix from the covariance between a previous-layer unit's output and the next-layer unit's input, and combining it with local differential estimation inside the units propagates the output error recursively along the computational graph, with no transposed-weight readout and no backward data path. The resulting gradient is empirically near-unbiased, and with local per-weight Adam updates it matches the final accuracy of backpropagation on simple regression tasks. With uniformly distributed noise, the local operations reduce to polynomials and comparators, making the whole system, learning rule included, well suited to digital circuits. Thus, in the NNN, noise is a resource not only for inference but also for reconstructing backpropagation.
Full-waveform inversion (FWI) estimates unknown parameters in the wave equation from limited boundary measurements. Recent advances in neural reparameterized FWI (NeurFWI) demonstrate that representing the parameters using a neural network can reduce the reliance on the high-quality initial model and wavefield data, at the cost of slow high-resolution convergence. However, its underlying theoretical mechanism remains unclear. In this study, we establish the neural sensitivity kernel (NSK) and the wave tangent kernel (WTK) to analyze their convergence behavior from both model and data domains. These theoretical frameworks show that the neural tangent kernel (NTK) induced by neural representation adaptively modulates the original sensitivity and wave tangent kernels. This modulation leads to several key outcomes, i.e., the spectral filtering effect, the gradient wavenumber modulation, and the wave frequency bias, connecting the convergence behavior of NeurFWI with the eigen-structures of NSK and WTK. Building on these insights, we propose several enhanced NeurFWI methods with tailored eigen-structures in NSK and WTK to improve inversion performances and efficiency. We numerically validate these theoretical claims and the proposed methods in seismic exploration, and firstly extend their application to medical imaging.
Temporal Predictive Coding provides a layer-local, parallelisable mechanism for learning in recurrent systems, making it an attractive candidate for online local learning on neuromorphic and edge hardware. However, its recurrent parameter update captures only local temporal relationships, neglecting the historic influence of parameters along the latent-state trajectory, and therefore struggles to assign credit over longer temporal horizons. This work combines for the first time Temporal Predictive Coding with Real-Time Recurrent Learning (tPC-RTRL), incorporating an online influence matrix that tracks this historic effect whilst preserving the spatial and temporal locality properties valued by neuromorphic implementations. Under explicit assumptions, we prove that tPC-RTRL recovers the gradients of backpropagation-through-time exactly. Empirically, a near-equivalence holds across several tasks of varying scale and complexity, including byte-level language modelling on WikiText-103 (tPC-RTRL vs. BPTT: 1.865 vs. 1.864 validation BPC), English--French translation on a CCMatrix subset (20.23 vs. 20.29 BLEU), and a realistic nanodrone system-identification benchmark (0.506m vs. 0.505m mean position error). Finally, we show that the iterative inference mechanism used during training can be reused at deployment time to incorporate intermittent state observations, halving final-position error relative to open-loop rollout on the nanodrone task (0.402m vs. 0.805m) and suggesting a path towards unifying learning and filtering within the same computational framework.