Observer-based neural networks for flow estimation and control

TL;DR

This work tackles real-time flow-state estimation under sparse, noisy sensing by introducing observer-based neural networks (NNOs) that generalize the Luenberger observer through neural network surrogates. The framework fuses a neural network surrogate model (NNSM) of the flow, an output-model NN, and a nonlinear NN observer (NNO) to produce dynamic state estimates that feed a controller NN (NNC) for closed-loop control. The approach is validated on three nonlinear systems: a modified Kuramoto-Sivashinsky equation, a confined cylinder flow, and turbulent boundary-layer data, demonstrating accurate state reconstruction and substantial stabilization when possible, even with reduced sensor counts and noisy measurements. Key findings include robustness to measurement noise and sampling-rate reductions, and improved estimation performance over direct super-resolution reconstructions by exploiting the predictive memory of the NNSM within the observer loop. The framework offers a feasible path toward real-time deployment in practical flows, with implications for sensor placement, actuation strategies, and hardware implementation (e.g., FPGA or microcontroller) to enable online state feedback control.

Abstract

Neural network observers (NNOs) are proposed for real-time estimation of fluid flows, addressing a key challenge in flow control: obtaining real-time flow states from a limited set of sparse and noisy sensor data. For this task, we propose a generalization of the classical Luenberger observer. In the present framework, the estimation loop is composed of subsystems modeled as neural networks (NNs). By combining flow information from selected probes and an NN surrogate model (NNSM) of the flow system, we train NNOs capable of fusing information to provide the best estimation of the states, that can in turn be fed back to an NN controller (NNC). The NNO capabilities are demonstrated for three nonlinear dynamical systems. First, a variation of the Kuramoto-Sivashinsky (KS) equation with control inputs is studied, where variables are sparsely probed. We show that the NNO is able to track states even when probes are contaminated with random noise or with sensors at insufficient sample rates to match the control time step. Then, a confined cylinder flow is investigated, where velocity signals along the cylinder wake are estimated by using a small set of wall pressure sensors. In both the KS and cylinder problems, we show that the estimated states can be used to enable closed-loop control, taking advantage of stabilizing NNCs. Finally, we present a legacy dataset of a turbulent boundary layer experiment, where convolutional NNs (CNNs) are employed to implement the models required for the estimation loop. We show that, by combining low-resolution noise-corrupted sensor data with an imperfect NNSM, it is possible to produce more accurate estimates, outperforming both the direct reconstructions via specialized super-resolution NNs and the direct model propagation from initial conditions.

Figures (35)

• Figure 1: Block diagram representing the discrete Luenberger observer for a linear dynamical system. The plant model is simulated in real time as the real system evolves. The correction signal $\bm{v}[k]$ is produced to rectify errors due to initial condition, plant imperfections and unmodelled disturbances.
• Figure 2: Block diagram representing a nonlinear generalisation of the discrete Luenberger observer. The colours are chosen to highlight the analogy with the classic approach for linear systems, depicted in figure \ref{['fig:loop_linear']}.
• Figure 3: Schematic of a single iteration of the observer training loop. The NNSM and output model weights are frozen and only the NNO ones are updated. The hat notation ($\hat{x}$ and $\hat{y}$) indicates estimated signals.
• Figure 4: Unrolled observer loop over a finite horizon of length $n_h$, with each neural network block representing the complete NNO loop.
• Figure 5: Finite-horizon observer training loop with controller (NNC) inclusion.