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Multi evolutional deep neural networks (Multi-EDNN)

Hadden Kim, Tamer A. Zaki

TL;DR

Multi-EDNN extends evolutional deep neural networks by introducing coupled EDNNs (C-EDNN) for state-wise, interdependent components and distributed EDNNs (D-EDNN) for spatial domain decomposition. By employing correction functions and flux corrections across interfaces, it maintains PDE consistency while enabling parallel training and smaller per-network models. The framework is demonstrated on linear advection, linear diffusion, Couette flow, and Taylor-Green vortices, achieving high accuracy and favorable scalability. This work provides a pathway toward scalable ML-based PDE solvers for large-scale, complex systems, with potential extensions to irregular geometries and adaptive sampling strategies.

Abstract

Evolutional deep neural networks (EDNN) solve partial differential equations (PDEs) by marching the network representation of the solution fields, using the governing equations. Use of a single network to solve coupled PDEs on large domains requires a large number of network parameters and incurs a significant computational cost. We introduce coupled EDNN (C-EDNN) to solve systems of PDEs by using independent networks for each state variable, which are only coupled through the governing equations. We also introduce distributed EDNN (D-EDNN) by spatially partitioning the global domain into several elements and assigning individual EDNNs to each element to solve the local evolution of the PDE. The networks then exchange the solution and fluxes at their interfaces, similar to flux-reconstruction methods, and ensure that the PDE dynamics are accurately preserved between neighboring elements. Together C-EDNN and D-EDNN form the general class of Multi-EDNN methods. We demonstrate these methods with aid of canonical problems including linear advection, the heat equation, and the compressible Navier-Stokes equations in Couette and Taylor-Green flows.

Multi evolutional deep neural networks (Multi-EDNN)

TL;DR

Multi-EDNN extends evolutional deep neural networks by introducing coupled EDNNs (C-EDNN) for state-wise, interdependent components and distributed EDNNs (D-EDNN) for spatial domain decomposition. By employing correction functions and flux corrections across interfaces, it maintains PDE consistency while enabling parallel training and smaller per-network models. The framework is demonstrated on linear advection, linear diffusion, Couette flow, and Taylor-Green vortices, achieving high accuracy and favorable scalability. This work provides a pathway toward scalable ML-based PDE solvers for large-scale, complex systems, with potential extensions to irregular geometries and adaptive sampling strategies.

Abstract

Evolutional deep neural networks (EDNN) solve partial differential equations (PDEs) by marching the network representation of the solution fields, using the governing equations. Use of a single network to solve coupled PDEs on large domains requires a large number of network parameters and incurs a significant computational cost. We introduce coupled EDNN (C-EDNN) to solve systems of PDEs by using independent networks for each state variable, which are only coupled through the governing equations. We also introduce distributed EDNN (D-EDNN) by spatially partitioning the global domain into several elements and assigning individual EDNNs to each element to solve the local evolution of the PDE. The networks then exchange the solution and fluxes at their interfaces, similar to flux-reconstruction methods, and ensure that the PDE dynamics are accurately preserved between neighboring elements. Together C-EDNN and D-EDNN form the general class of Multi-EDNN methods. We demonstrate these methods with aid of canonical problems including linear advection, the heat equation, and the compressible Navier-Stokes equations in Couette and Taylor-Green flows.
Paper Structure (16 sections, 70 equations, 8 figures)

This paper contains 16 sections, 70 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem setup and preliminaries
  3. Formulation of Multi Evolutional Deep Neural Networks (Multi-EDNN)
  4. Coupled EDNNs
  5. Distributed EDNNs
  6. Correction functions
  7. Boundary conditions
  8. Numerical experiments
  9. Couette flow
  10. Linear advection
  11. Linear diffusion
  12. Taylor-Green vortices
  13. Conclusions
  14. Adaptive sampling
  15. Riemann solver choices
  16. ...and 1 more sections

Figures (8)

  • Figure 1: Schematic of (a) original EDNN solving a system of PDEs on the entire spatial domain and (b) Multi-EDNN solving coupled differential equations using distributed networks on four subdomains. In Multi-EDNN, each subdomain has two coupled networks that solve the system of equations locally, and that exchange information at the sub-domain interfaces with their neighbors.
  • Figure 2: One-dimensional (a) correction functions and (b) their derivatives for left boundary. A fifth-order polynomial $DG_5$ (black dashed) and the monomial $\mathcal{M}_{15,0.1}$ (red solid).
  • Figure 3: C-EDNN solution of Couette flow. Analytical (grey) and C-EDNN prediction (colored) of steady-state (a) $\rho u _1$ and (b) temperature for Mach numbers $Ma = \{0.2, 0.5, 1.2, 2.0\}$. (c) Analytical steady-state (grey) and C-EDNN prediction (colored) of $\rho u _1$ for $Ma = 4$, $Re = 100$ at times $t= \{0, 0.1, 1, 10\}$.
  • Figure 4: Predictions of 1D advection equation for $\ell=2$ without correction. Analytical solution (grey) and D-EDNN prediction (colored) at time (a) $t=0.625$, (b) $t=1$, (b) $t=1.375$, and (d) $t=1.625$
  • Figure 5: Predictions of 1D advection equation for $\ell=21$ with correction. Analytical solution (grey) and D-EDNN prediction (colored) at time (a) $t=0.625$, (b) $t=10$, and (c) $t=20$. (d) Instantaneous error $\varepsilon$ over time.
  • ...and 3 more figures