Research ArticleAPPLIED PHYSICS

Wave physics as an analog recurrent neural network

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Science Advances  20 Dec 2019:
Vol. 5, no. 12, eaay6946
DOI: 10.1126/sciadv.aay6946
  • Fig. 1 Conceptual comparison of a standard RNN and a wave-based physcal system.

    (A) Diagram of an RNN cell operating on a discrete input sequence and producing a discrete output sequence. (B) Internal components of the RNN cell, consisting of trainable dense matrices W(h), W(x), and W(y). Activation functions for the hidden state and output are represented by σ(h) and σ(y), respectively. (C) Diagram of the directed graph of the RNN cell. (D) Diagram of a recurrent representation of a continuous physical system operating on a continuous input sequence and producing a continuous output sequence. (E) Internal components of the recurrence relation for the wave equation when discretized using finite differences. (F) Diagram of the directed graph of discrete time steps of the continuous physical system and illustration of how a wave disturbance propagates within the domain.

  • Fig. 2 Schematic of the vowel recognition setup and the training procedure.

    (A) Raw audio waveforms of spoken vowel samples from three classes. (B) Layout of the vowel recognition system. Vowel samples are independently injected at the source, located at the left of the domain, and propagate through the center region, indicated in green, where a material distribution is optimized during training. The dark gray region represents an absorbing boundary layer. (C) For classification, the time-integrated power at each probe is measured and normalized to be interpreted as a probability distribution over the vowel classes. (D) Using automatic differentiation, the gradient of the loss function with respect to the density of material in the green region is computed. The material density is updated iteratively, using gradient-based stochastic optimization techniques until convergence.

  • Fig. 3 Vowel recognition training results.

    Confusion matrix over the training and testing datasets for the initial structure (A and B) and final structure (C and D), indicating the percentage of correctly (diagonal) and incorrectly (off-diagonal) predicted vowels. Cross-validated training results showing the mean (solid line) and SD (shaded region) of the (E) cross-entropy loss and (F) prediction accuracy over 30 training epochs and five folds of the dataset, which consists of a total of 279 total vowel samples of male and female speakers. (G to I) The time-integrated intensity distribution for a randomly selected input (G) ae vowel, (H) ei vowel, and (I) iy vowel.

  • Fig. 4 Frequency content of the vowel classes.

    The plotted quantity is the mean energy spectrum for the ae, ei, and iy vowel classes. a.u., arbitrary units.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/12/eaay6946/DC1

    Section S1. Derivation of the wave equation update relationship

    Section S2. Realistic physical platforms and nonlinearities

    Section S3. Input and output connection matrices

    Section S4. Comparison of wave RNN and conventional RNN

    Section S5. Binarization of the wave speed distribution

    Fig. S1. Saturable absorption response.

    Fig. S2. Cross-validated training results for an RNN with a saturable absorption nonlinearity.

    Table S1. Comparison of a scalar wave model and a conventional RNN on a vowel recognition task.

    References (3543)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Derivation of the wave equation update relationship
    • Section S2. Realistic physical platforms and nonlinearities
    • Section S3. Input and output connection matrices
    • Section S4. Comparison of wave RNN and conventional RNN
    • Section S5. Binarization of the wave speed distribution
    • Fig. S1. Saturable absorption response.
    • Fig. S2. Cross-validated training results for an RNN with a saturable absorption nonlinearity.
    • Table S1. Comparison of a scalar wave model and a conventional RNN on a vowel recognition task.
    • References (3543)

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