APEX: Amplitude Anchors and Phase Priors for Target-Scarce Higher-Frequency Wave Prediction

Technical Summary: APEX

Problem Statement
Learning-based surrogates, particularly neural operators, have demonstrated efficacy in predicting wave fields within observed frequency regimes. However, predicting higher-frequency wave fields under scarce target supervision remains a significant challenge. In many physical domains (e.g., electromagnetics, acoustics), higher-frequency data is substantially more expensive to simulate or measure due to requirements for finer discretization and stricter numerical control. Consequently, practitioners often possess abundant lower-frequency data but only limited supervision in the higher-frequency target regime.

Existing strategies, such as direct lower-to-higher extrapolation or fine-tuning low-frequency models on limited high-frequency data, often fail in highly oscillatory regimes. The paper identifies that the core difficulty is not merely a uniform increase in error, but an inherent asymmetry in cross-frequency transferability: while coarse amplitude structures remain relatively stable across frequencies, phase-sensitive oscillatory structures deteriorate rapidly as the frequency gap increases.

Methodology: APEX
To address this asymmetry, the authors propose APEX (Amplitude-anchored and Phase-prior-guided Enhancement from eXtrapolated coarse predictions). The framework decomposes the higher-frequency prediction task into two distinct stages:

1. Coarse Amplitude Anchor:
   A neural operator (specifically a Fourier Neural Operator, FNO) is trained exclusively on abundant lower-frequency data and then frozen. When queried at a target higher frequency, this operator provides a coarse prediction. The framework extracts only the log-amplitude of this prediction, treating it as a transferable structural anchor. The phase information from this extrapolation is discarded as it is deemed unreliable in the higher-frequency regime.

2. Green's-Function-Inspired Phase Prior:
   To recover the missing oscillatory detail, APEX introduces an explicit phase prior. This prior is a lightweight, analytic surrogate based on a geometric propagation view. It approximates the accumulated phase along dominant propagation paths using the formula:
   $$G_{prior}(r, r_s; e, \nu) = \sum_{m=1}^{M} a_m \exp(i \kappa_{ref}(e, \nu) L_m(r, r_s))$$
   where $L_m$ is the geometric path length, $\kappa_{ref}$ is a domain-adaptive reference phase coefficient, and $M$ represents the number of retained dominant paths. This provides a source- and geometry-aware oscillatory scaffold without attempting to reproduce the full complex field.

3. Conditional Flow-Matching Enhancer:
   A generative model based on Conditional Flow Matching (CFM) is employed to reconstruct the target higher-frequency field. The model is conditioned on three components:

   • The coarse amplitude anchor ($a_{coarse}$).
   • The sine and cosine of the phase prior ($\sin \phi_{base}, \cos \phi_{base}$).
   • Fourier feature embeddings of the target frequency.

   The enhancer learns to map a noise distribution to the target complex field (represented as log-amplitude and sine/cosine phase) by minimizing a flow-matching loss, effectively recovering the high-frequency oscillatory details that the coarse anchor and phase prior alone cannot fully capture.

Key Contributions

• Identification of Transfer Asymmetry: The paper empirically and theoretically demonstrates that in complex wave fields, coarse amplitude structure is significantly more transferable across frequencies than phase-sensitive oscillatory structure. This is formalized through a regional error decomposition showing that phase errors scale quadratically with frequency ($\nu^2$), whereas amplitude errors do not.
• Decomposed Framework (APEX): A novel architecture that explicitly separates the prediction task into reusing transferable coarse structure (amplitude) and separately recovering missing oscillatory detail (phase) guided by physics-inspired priors.
• Performance under Scarcity: The method is specifically designed for scenarios with scarce target-frequency supervision, outperforming direct extrapolation, target-adapted operators, and joint generative baselines.

Experimental Results
APEX was evaluated on three benchmarks: SimpleWave (a two-path wave simulator), Helmholtz (heterogeneous frequency-domain wave equation), and Maxwell (full-wave electromagnetic simulations). The evaluation focused on extrapolation to frequencies 20%, 50%, and 100% higher than the maximum training frequency.

• Quantitative Performance: Across all benchmarks, APEX consistently achieved the lowest relative complex $H^1$ error and the highest Amplitude-Weighted Phase Coherence (AWPC). For instance, on the Helmholtz benchmark at HF100 (100% extrapolation), APEX achieved an $H^1$ error of 0.634 and AWPC of 0.815, significantly outperforming the best baseline (FNO-Joint: $H^1$ 0.899, AWPC 0.737).
• Qualitative Analysis: Visual comparisons show that while direct extrapolation preserves some coarse amplitude but fails on phase alignment, and joint generative models produce plausible but misaligned structures, APEX successfully recovers both the amplitude localization and the coherent phase patterns of the ground truth.
• Ablation Studies: Removing either the phase prior or the coarse anchor resulted in significant performance degradation, confirming that both components are necessary and complementary. The coarse anchor provides global organization, while the phase prior guides fine oscillatory recovery.

Significance and Claims
The paper claims that reliable higher-frequency prediction of oscillatory wave fields should not rely on the direct end-to-end transfer of the full complex field. Instead, it argues for a strategy that explicitly reuses transferable coarse structure (amplitude) while separately recovering the missing oscillatory detail using physics-guided priors.

The authors position APEX as a solution to the "scarce supervision" problem in high-frequency wave modeling, offering a way to leverage abundant low-frequency data without requiring extensive high-frequency datasets for training. The work suggests that understanding the specific components of a field that degrade under frequency shifts is crucial for designing generalizable surrogate models.

Limitations
The authors acknowledge that the inference cost of APEX is higher than a single forward pass of a neural operator due to the iterative sampling required by the flow-matching enhancer. Additionally, the phase prior is a simplified approximation that does not fully model complex multipath effects or strong dispersion, though it serves as an effective lightweight scaffold. The current implementation uses FNO as the backbone, though the amplitude-anchor concept is theoretically backbone-agnostic.