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Learning Vocal-Tract Area and Radiation with a Physics-Informed Webster Model

Minhui Lu, Joshua D. Reiss

Abstract

We present a physics-informed voiced backend renderer for singing-voice synthesis. Given synthetic single-channel audio and a fund-amental--frequency trajectory, we train a time-domain Webster model as a physics-informed neural network to estimate an interpretable vocal-tract area function and an open-end radiation coefficient. Training enforces partial differential equation and boundary consistency; a lightweight DDSP path is used only to stabilize learning, while inference is purely physics-based. On sustained vowels (/a/, /i/, /u/), parameters rendered by an independent finite-difference time-domain Webster solver reproduce spectral envelopes competitively with a compact DDSP baseline and remain stable under changes in discretization, moderate source variations, and about ten percent pitch shifts. The in-graph waveform remains breathier than the reference, motivating periodicity-aware objectives and explicit glottal priors in future work.

Learning Vocal-Tract Area and Radiation with a Physics-Informed Webster Model

Abstract

We present a physics-informed voiced backend renderer for singing-voice synthesis. Given synthetic single-channel audio and a fund-amental--frequency trajectory, we train a time-domain Webster model as a physics-informed neural network to estimate an interpretable vocal-tract area function and an open-end radiation coefficient. Training enforces partial differential equation and boundary consistency; a lightweight DDSP path is used only to stabilize learning, while inference is purely physics-based. On sustained vowels (/a/, /i/, /u/), parameters rendered by an independent finite-difference time-domain Webster solver reproduce spectral envelopes competitively with a compact DDSP baseline and remain stable under changes in discretization, moderate source variations, and about ten percent pitch shifts. The in-graph waveform remains breathier than the reference, motivating periodicity-aware objectives and explicit glottal priors in future work.
Paper Structure (15 sections, 7 equations, 2 figures, 4 tables)

This paper contains 15 sections, 7 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Physics-informed Voiced Renderer
  3. Governing equations and boundary conditions
  4. Physics losses
  5. Differentiable audio and probes
  6. Auxiliary DDSP renderer (training only)
  7. Overall objective
  8. Training and Evaluation Protocol
  9. Results
  10. Post-render validation: recovered controls transfer beyond the training graph
  11. The periodicity gap: in-graph rendering is systematically more aperiodic
  12. Learned $A(x)$ and $\zeta$: transferable controls but not uniquely identifiable under steady vowels
  13. Robustness to controlled mismatches
  14. Audio examples and qualitative observations
  15. Conclusion

Figures (2)

  • Figure 1: Overview of the physics-informed voiced renderer. DualNet predicts $(\psi,\hat{A},\hat{\zeta})$ and a differentiable Webster rendering path produces $\hat{y}(t)$ for reference-based losses during training (inference is physics-only). Solid arrows denote forward signal flow in the renderer; dashed arrows denote training-only loss/backprop connections (e.g., using $y(t)$), which are removed at inference. For solver-independent evaluation (not shown), $(\hat{A},\hat{\zeta})$ are exported to an independent FDTD--Webster solver for post-render assessment.
  • Figure 2: Recovered area functions $\hat{A}(x)$ (normalised units). Here $x$ increases from the glottis $(0)$ to the lips $(1)$. The solutions capture broad vowel-dependent trends (e.g., anterior constriction for /i/ and a narrower mouth end for /u/), while fine-scale details remain ambiguous under single-channel steady voiced supervision.