Simulating Three-dimensional Turbulence with Physics-informed Neural Networks

TL;DR

This paper demonstrates that physics-informed neural networks can learn fully developed turbulence in 2D and 3D without data, by integrating architectural and training innovations. The authors introduce PirateNet and RWF to enable deep, multiscale PINNs, exact periodic BC enforcement, causal training, adaptive loss weighting, a SOAP optimizer, and time-marching with transfer learning. They validate on three benchmarks—2D Kolmogorov flow, 3D Taylor–Green vortex, and 3D turbulent channel flow—and show accurate statistics such as energy spectra, kinetic energy, enstrophy, and Reynolds stresses, approaching high-fidelity DNS performance. While not yet beating spectral solvers in efficiency, the approach provides a proof of concept for mesh-free, continuous turbulence modeling and lays groundwork for hybrid data-assimilation and inverse problems.

Abstract

Turbulent fluid flows are among the most computationally demanding problems in science, requiring enormous computational resources that become prohibitive at high flow speeds. Physics-informed neural networks (PINNs) represent a radically different approach that trains neural networks directly from physical equations rather than data, offering the potential for continuous, mesh-free solutions. Here we show that appropriately designed PINNs can successfully simulate fully turbulent flows in both two and three dimensions, directly learning solutions to the fundamental fluid equations without traditional computational grids or training data. Our approach combines several algorithmic innovations including adaptive network architectures, causal training, and advanced optimization methods to overcome the inherent challenges of learning chaotic dynamics. Through rigorous validation on challenging turbulence problems, we demonstrate that PINNs accurately reproduce key flow statistics including energy spectra, kinetic energy, enstrophy, and Reynolds stresses. Our results demonstrate that neural equation solvers can handle complex chaotic systems, opening new possibilities for continuous turbulence modeling that transcends traditional computational limitations.

Figures (19)

• Figure 1: Physics-informed neural network framework for solving the incompressible Navier-Stokes equations. The PINN model takes spatiotemporal coordinates as input and predicts the velocity and pressure fields as $\mathbf{u}_\theta = (u_\theta, v_\theta, w_\theta, p_\theta)$, with $\theta$ denoting the network parameters. The network is trained by minimizing a composite loss that penalizes violations of PDE constraints and initial/boundary conditions, with PDE residuals computed via automatic differentiation. The framework integrates several key innovations: (a) a physics-informed residual adaptive network (PirateNet) that mitigates spectral bias and supports deep architectures; (b) a causal loss formulation that enforces temporal causality during training; (c) a self-adaptive weighting strategy that dynamically balances contributions from different loss terms; (d) a quasi-second-order optimizer (SOAP) that addresses conflicting gradient directions; and (e) a time-marching scheme with transfer learning, where the trainable parameters of models for later time windows are initialized using the converged model parameters for earlier time windows. These components collectively enable robust and efficient training of large-scale PINNs, making them viable for simulating high Reynolds number turbulent flows.
• Figure 2: 2D Turbulent Kolmogorov Flow ($\text{Re}=10^6$). (a) Vorticity snapshots at selected time instances comparing predictions with reference DNS, computed using a pseudo-spectral method on a $2048 \times 2048$ uniform grid. The predicted fields closely match the DNS, accurately capturing fine-scale vortical structures and their temporal evolution. (b) Temporal evolution of the relative $L^2$ velocity error (spatially averaged), demonstrating sustained accuracy over the time domain. (c–d) Comparison of kinetic energy and enstrophy evolution between PINN predictions and DNS results, showing good agreement in the preservation of key physical quantities. (e) Comparison of the energy spectrum at the final time between PINN predictions and DNS, illustrating accurate recovery of both the inverse energy cascade ($k^{-5/3}$) and the forward enstrophy cascade ($k^{-3}$).
• Figure 3: Taylor-Green Vortex (Re=1600). (a) Evolution of the iso-surfaces of the Q-criterion $(Q=0.1)$ at different time snapshots, predicted by PINNs and colored by the non-dimensional velocity magnitude. (b–c) Temporal evolution of spatially averaged kinetic energy and enstrophy, comparing PINN predictions against a pseudo-spectral DNS (resolution $512^3$) and 8th-order finite difference solvers at various resolutions ($64^3$–$512^3$). The PINN achieves accuracy comparable to high-order solvers at moderate resolution and captures key dynamical features of the flow. (d) Comparison of the iso-contours of the dimensionless vorticity norm on the periodic face $x=-\pi$ at $t=8$.
• Figure 4: Turbulent channel flow ($\text{Re}_\tau \sim 395$). (a) Instantaneous snapshot of the predicted velocity norm, illustrating characteristic near-wall streaks and large-scale turbulent structures. (b) Comparison of the mean streamwise and spanwise velocity profile and pressrure profile between PINN predictions and DNS data, normalized in wall units. (c) Comparison of Reynolds stress components ($\overline{u^{\prime}u^{\prime}}^{+}$, $\overline{v^{\prime}v^{\prime}}^{+}$, $\overline{w^{\prime}w^{\prime}}^{+}$, and $\overline{u^{\prime} v^{\prime}}^{+}$) between PINN predictions and DNS data, normalized by the respective friction velocity $u_\tau$.
• Figure 5: 2D Turbulent Kolmogorov flow. Comparison of the predicted velocity $u$ against the corresponding numerical reference at different time snapshots.