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Physics-Aware Decoding for Communication Channels Governed by Partial Differential Equations

Tadashi Wadayama, Koji Igarashi, Takumi Takahashi

TL;DR

The paper tackles decoding over channels governed by partial differential equations, where conventional decoders ignore the underlying physics. It introduces physics-aware decoding by integrating gradient-flow decoding with differentiable PDE solvers, enabling gradient information from the physics model to guide error correction. Across heat equation and NLSE channels, the proposed approach improves decoding performance and demonstrates a new paradigm for physics-informed signal processing with end-to-end differentiability via automatic differentiation. A primary challenge is the computational overhead from the double-loop structure, suggesting future work on efficiency and broader applications in signal detection and recovery.

Abstract

Digital communication systems inherently operate through physical media governed by partial differential equations (PDEs). In this paper, we introduce a physics-aware decoding framework that integrates gradient descent-based error correcting algorithms with PDE-based channel modeling using differentiable PDE solvers. At the core of our approach is gradient flow decoding, which harnesses gradient information directly from the PDE solver to guide the decoding process. We validate our method through numerical experiments on both the heat equation and the nonlinear Schrödinger equation (NLSE), demonstrating significant improvements in decoding performance. The implications of this work extend beyond decoding applications, establishing a new paradigm for physics-aware signal processing that shows promise for various signal detection and signal recovery tasks.

Physics-Aware Decoding for Communication Channels Governed by Partial Differential Equations

TL;DR

The paper tackles decoding over channels governed by partial differential equations, where conventional decoders ignore the underlying physics. It introduces physics-aware decoding by integrating gradient-flow decoding with differentiable PDE solvers, enabling gradient information from the physics model to guide error correction. Across heat equation and NLSE channels, the proposed approach improves decoding performance and demonstrates a new paradigm for physics-informed signal processing with end-to-end differentiability via automatic differentiation. A primary challenge is the computational overhead from the double-loop structure, suggesting future work on efficiency and broader applications in signal detection and recovery.

Abstract

Digital communication systems inherently operate through physical media governed by partial differential equations (PDEs). In this paper, we introduce a physics-aware decoding framework that integrates gradient descent-based error correcting algorithms with PDE-based channel modeling using differentiable PDE solvers. At the core of our approach is gradient flow decoding, which harnesses gradient information directly from the PDE solver to guide the decoding process. We validate our method through numerical experiments on both the heat equation and the nonlinear Schrödinger equation (NLSE), demonstrating significant improvements in decoding performance. The implications of this work extend beyond decoding applications, establishing a new paradigm for physics-aware signal processing that shows promise for various signal detection and signal recovery tasks.
Paper Structure (17 sections, 23 equations, 5 figures, 1 algorithm)

This paper contains 17 sections, 23 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Gradient Flow (GF) Decoding
  5. Finite Difference Methods (FDM)
  6. State Split Fourier Method (SSFM)
  7. Decoding Problem for PDE Channel
  8. Overview of Channel Model
  9. Details of Channel Model
  10. Decoding Problem
  11. Proposed Method
  12. Overview of Physics-Aware Decoding
  13. Details of Physics-Aware Decoding
  14. Numerical Experiments
  15. Heat Equation
  16. ...and 2 more sections

Figures (5)

  • Figure 1: A PDE channel defined by a heat PDE with $\lambda = 0.2$. A bipolar vector $\bm s = (+1,+1,-1,-1,+1,-1,+1)$ is used as the input vector and Gaussian-shaped pulses are used for generating input waveform.
  • Figure 2: Block diagram of physics-aware decoding.
  • Figure 3: Received waveform and estimated output waveform $\bm u^{(N_t)}$ in a decoding process.
  • Figure 4: BER performance of the proposed algorithm for the PDE channel governed by the heat PDE. The (31,15) BCH code is used. The error curve of peak detection method is also included as the baseline.
  • Figure 5: BER performance of the proposed algorithm for the PDE channel governed by the NLSE (\ref{['NLE']}). (15,7) BCH code is used. Conventional BP method is used as a baseline.