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Physics-informed Neural-operator Predictive Control for Drag Reduction in Turbulent Flows

Zelin Zhao, Zongyi Li, Kimia Hassibi, Kamyar Azizzadenesheli, Junchi Yan, H. Jane Bae, Di Zhou, Anima Anandkumar

TL;DR

The paper tackles drag reduction in turbulent wall-bounded flows by introducing PINO-PC, a model-based reinforcement learning framework that learns both an observer (PINO) and a policy (FNO-based) in discretization-invariant function spaces. By combining physics-informed losses (PDE constraints) with neural-operator models, PINO-PC achieves superior drag reduction and robust generalization to unseen Reynolds numbers, outperforming model-free RL and traditional opposition controls. Key contributions include the integration of PINO for flow prediction, MFN-conditioned Reynolds-number generalization, and a predictive-control objective linking kinetic energy and actuation costs to drag reduction. The approach promises practical gains for efficient turbulence control by enabling online adaptation and transfer across scales, with strong empirical performance in DNS tests up to ${Re_b}=15{,}000$ and unseen flow regimes.

Abstract

Assessing turbulence control effects for wall friction numerically is a significant challenge since it requires expensive simulations of turbulent fluid dynamics. We instead propose an efficient deep reinforcement learning (RL) framework for modeling and control of turbulent flows. It is model-based RL for predictive control (PC), where both the policy and the observer models for turbulence control are learned jointly using Physics Informed Neural Operators (PINO), which are discretization invariant and can capture fine scales in turbulent flows accurately. Our PINO-PC outperforms prior model-free reinforcement learning methods in various challenging scenarios where the flows are of high Reynolds numbers and unseen, i.e., not provided during model training. We find that PINO-PC achieves a drag reduction of 39.0\% under a bulk-velocity Reynolds number of 15,000, outperforming previous fluid control methods by more than 32\%.

Physics-informed Neural-operator Predictive Control for Drag Reduction in Turbulent Flows

TL;DR

The paper tackles drag reduction in turbulent wall-bounded flows by introducing PINO-PC, a model-based reinforcement learning framework that learns both an observer (PINO) and a policy (FNO-based) in discretization-invariant function spaces. By combining physics-informed losses (PDE constraints) with neural-operator models, PINO-PC achieves superior drag reduction and robust generalization to unseen Reynolds numbers, outperforming model-free RL and traditional opposition controls. Key contributions include the integration of PINO for flow prediction, MFN-conditioned Reynolds-number generalization, and a predictive-control objective linking kinetic energy and actuation costs to drag reduction. The approach promises practical gains for efficient turbulence control by enabling online adaptation and transfer across scales, with strong empirical performance in DNS tests up to and unseen flow regimes.

Abstract

Assessing turbulence control effects for wall friction numerically is a significant challenge since it requires expensive simulations of turbulent fluid dynamics. We instead propose an efficient deep reinforcement learning (RL) framework for modeling and control of turbulent flows. It is model-based RL for predictive control (PC), where both the policy and the observer models for turbulence control are learned jointly using Physics Informed Neural Operators (PINO), which are discretization invariant and can capture fine scales in turbulent flows accurately. Our PINO-PC outperforms prior model-free reinforcement learning methods in various challenging scenarios where the flows are of high Reynolds numbers and unseen, i.e., not provided during model training. We find that PINO-PC achieves a drag reduction of 39.0\% under a bulk-velocity Reynolds number of 15,000, outperforming previous fluid control methods by more than 32\%.

Paper Structure

This paper contains 30 sections, 1 theorem, 20 equations, 9 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results
  3. Methods
  4. Problem setting
  5. The governing equation
  6. The solver
  7. Control setups
  8. Computational domains
  9. The Reynolds numbers
  10. Algorithm outline of PINO-PC
  11. The policy model MP
  12. The FNO encoder of the policy model
  13. Conditioning on the Reynolds number
  14. The FNO decoder of the policy model
  15. The policy loss
  16. ...and 15 more sections

Key Result

Theorem 1

Suppose that the MDP satisfies the following regularization conditions then $\nabla_\theta \mu_\theta(x)$ and $D_a(Q^\mu(x, a))$ exist and the deterministic policy gradient is given as:

Figures (9)

  • Figure 1: (a): Channel flow moving along the streamwise direction $x$, with the control applied at the wall via suction or blowing. (b): Overall schematic of PINO-PC. PINO-PC consists of two neural operator components: a Physics-Informed Neural Operator (PINO PINO) that serves as the observer model for flow prediction, and a Fourier Neural Operator (FNO FNO) that acts as the policy model for control action generation. The controller takes pressure observations as input, performs predictive control, and applies the control to the turbulent flow environment. The PINO observer integrates both data and physics-based losses during training to learn accurate flow dynamics, while the FNO policy model optimizes control actions to minimize drag. Data is collected from the turbulent flow environment and is used to train the model.
  • Figure 2: Drag reduction curves comparing several flow control methods: MP-CNNMLoppositionControl, Local suboptimal lee1998suboptimal, DDPGDeepRLchannel and PINO-PC (ours). The $x$-axis denotes the non-dimensional timestep, and the $y$-axis denotes the drag reduction rate (DR) related to the uncontrolled case. The control beginning time is indicated via a red circle.
  • Figure 3: Time averaged statistics of the uncontrolled flow and PINO-PC (after control) in the full channel flow case. (a) The mean stream-wise velocity profile. (b, c, d) Stream-wise, wall-normal, and span-wise r.m.s. velocity fluctuations in the wall-normal direction.
  • Figure 4: Joint probability density function of the streamwise and wall-normal velocity fluctuations at $y^+=15$ for (a) the uncontrolled full-channel flow and (b) the PINO-PC (with control) full-channel flow. The contour lines denote 20%, 40%, 60%, and 80% of the maximum probability density values.
  • Figure 5: Premultiplied energy spectra of the streamwise velocity fluctuations $u'$ for (a, c) the uncontrolled full-channel flow and (b, d) the PINO-PC (with control) full-channel flow as a functions of wall-distance and wavelengths.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 1: Deterministic policy gradient theorem for neural operators
  • proof