Organizations: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China · School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, China · School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610000, China
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.
Periodic Implicit Neural Representations (INRs) such as SIREN and FINER assign every neuron, the same global frequency, spending the representational budget inefficiently when local signal content varies. We introduce FiRe (Frequency Reparameterization), that accelerates optimization by reparameterizing per-neuron frequency of periodic INRs without changing their underlying activation function. FiRe gives each neuron a bounded, input-dependent frequency via a separate low-rank gating path and is applicable to any periodic activation function. The gate acts as an implicit preconditioner that improves optimization conditioning at initialization via the Neural Tangent Kernel (NTK). This better-conditioned initialization makes optimization converge faster, and the high-frequency content of the reconstruction tracks the target more closely at a fixed computational budget. On 2D image fitting, FiRe increases PSNR over a parameter-matched baseline (up to +1 dB at short training budgets), with gains that vary with resolution and diminish at full convergence. We characterize how performance depends on resolution, rank, and training budget, and give an NTK account that predicts these trends.
Full waveform inversion (FWI) reconstructs heterogeneous material properties from receiver data but remains computationally demanding. Physics-informed neural networks (PINNs) and their domain-decomposed variants (FBPINNs) offer a mesh-free alternative but face convergence challenges when representing complex velocity fields. We present a hybrid quantum-classical FBPINN for acoustic FWI, bringing together quantum computing and classical machine learning, in which the decomposed wavefield network and the global velocity network are implemented as classical-to-quantum pipelines terminating in parameterized quantum circuits (PQCs). The PQCs are realized as differentiable JAX statevector simulators, enabling end-to-end automatic differentiation through the classical PINN, the quantum circuit, and the physics-informed loss. On a geophysical anomaly benchmark, the quantum hybrid reaches a lower L1 velocity error than the primary classical FBPINN baseline in approximately 8x fewer training iterations, despite using approximately 33% fewer trainable parameters, and it outperforms all 15 classical hyperparameter variants tested. A second benchmark (checkerboard) demonstrates the generality of the inversion pipeline, confirming that the quantum hybrid architecture can recover structured spatial variations beyond the localized anomaly benchmark. Our framework is broadly applicable to wave-based inverse problems beyond geophysics, including medical ultrasound tomography and non-destructive evaluation.
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.