Machine Learning Accelerated PDE Backstepping Observers

Historically, the study of observability of PDEs goes back to at least the 1960s, whereas infinite-dimensional Kalman filters emerge later in the early 1970s. However, the associated operator Riccati equations in these optimal state estimators are nonlinear infinite-dimensional equations, exacting a high price in not only numerical computation but also in the numerical analysis skills demanded from the user of such a state estimator. To address these challenges, a PDE backstepping approach for observer design was introduced in the 2005 paper  [1]. Backstepping provides three properties that match or exceed the properties of other PDE observer design methods: (1) convergence to the state of the actual PDE system is guaranteed; (2) convergence rate can be assigned (made arbitrarily rapid) using the design parameters in the backstepping approach; (3) the gain functions in the backstepping approach can be explicitly computed with arbitrary accuracy because these gain functions satisfy linear integral equations of Volterra’s second kind, rather than being governed by operator Riccati equations or nonlinear PDEs.

The PDE backstepping observers [1] are of what is commonly referred to as the “Luenberger structure,” consisting of a copy of the PDE model and an output estimation error term, multiplied by a gain function that depends on the spatial variable(s). The PDE backstepping design provides the gain function, without the need to solve any PDE in real time. However, the observer itself is a PDE whose input is the measured output of the original control system. Conducting state estimation in real time, in a fashion that requires a rapid solution to a PDE, may be challenging. But rapid estimation is necessary for many applications, including those where the state estimate from the PDE observer is used in a state feedback controller for stabilization, as in [1].

It is for this reason that we are interested in methods that eliminate the need for numerically solving the observer PDEs in real time. We pursue this in the present paper using an FNO approximation of the observer PDEs.

Literature Overview on PDE Backstepping Observers.¹¹1Even though this literature review may appear as an indulgence in self-citation, the objective is to provide the reader with helpful information on the nearly 20-year history of the subject of PDE backstepping observers. Following the introduction of the PDE backstepping observer design approach in [1] for parabolic 1-D PDEs, numerous extensions of this approach to hyperbolic and other PDE structures have followed. Extensions to wave PDEs were introduced in [2], and to beam models arising in atomic force microscopy in [3]. An adaptive observer for a single first-order hyperbolic PDE is reported in [4]. Designs for coupled systems of first-order hyperbolic PDEs are given in [5], including an experimental application to off-shore oil drilling in [6]. Generalizations from observers for PDEs to backstepping observers for ODE-PDE-ODE “sandwich” systems are given in [7] for the parabolic class and in [8] for the hyperbolic class. Observer extensions to less standard PDE classes are given in [9] for the Schrödinger PDE, in [10] for the Ginzburg-Landau PDE model of vortex shedding, and in [11] for the nonlinear Burgers PDE. Observers for PDE systems formulated in higher dimensions in polar and spherical coordinates are given for the annulus geometry in [12] for thermal convection and in [13] for PDEs on arbitrary-dimensional balls. All these results ensure exponential convergence of the state estimate. A design for “prescribed-time” observer convergence (independent of the distance between the system initial condition and the estimate initial condition) is given in [14].In addition to the applications mentioned above [3, 6, 11], backstepping PDE observers for the following applications have also been developed: for Lithium-ion batteries in [15], for the Stefan PDE-ODE model of phase transition arising in additive manufacturing [16], and Arctic sea ice estimation [17], for the Navier-Stokes and magnetohydrodynamic models of turbulent fluid flows in [18], and for traffic flows in [19].

Contributions. We use the Fourier neural operator as a surrogate model to accelerate the PDE backstepping observers. We propose two neural observer formulations: feedforward and recurrent. The former is similar to the smoothing process: the neural observer is given the observations on a time interval to estimate the solution on the interval. The latter is similar to the smoothing filtering process: the neural observer is given the observation at one time step and to predict for one time step. The feedforward neural observer gets higher accuracy whereas the recurrent neural observer holds more promise for high-frequency sample-data control.

The paper’s numerical results are given in Section IV, where three examples of accelerated PDE backstepping observer implementations are presented: exponential and prescribed-time observers for certain 1-D reaction-diffusion PDE examples, followed by a coupled first-order hyperbolic PDE model of stop-and-go oscillations in congested traffic. In all examples, the FNO observers, once trained over a few hours using a large set of initial conditions, perform real-time calculations up to 1000x faster than the basic PDE solvers.

No theorems are stated in the paper since, for the three experimental results in Section IV, the observer convergence theorems are provided, respectively, in [1, Theorem 8], [14, Proposition 5.3], [19, Theorem 3].

When implementing backstepping observers, numerical solvers such as finite difference methods and finite element methods are typically used to solve the boundary value problems. These conventional numerical solvers may be subjective to stability constraints (depending on the class of PDE) and therefore smaller time steps need to be used, which is computationally expensive. Recently, machine learning (ML) methods have shown promise in solving PDEs. Examples include impressive speedups in complex systems that model systems governing weather forecasting [20] and producing numerical solutions to previously unsolvable problems such as high-dimensional PDEs [21]. Inspired by these recent advances, we propose to use the data-driven neural operator as a surrogate solver for solving backstepping PDE observers. The neural operator solver shows orders of magnitude of speedup, making it more amenable to real-time applications.

ML methods for solving PDE frame the problem as a statistical inference task. In the standard supervised learning task, a dataset containing inputs (boundary conditions or initial conditions) and outputs (solution functions) are given, and the task is to approximate the solution operator between the input and output function spaces. Once such solution operator is learned, it can directly evaluate for any new input queries without solving any PDEs. As a result, the data-driven solver can be orders of magnitude faster compared to conventional solvers.

The model design is one of the most critical questions in ML. An ideal model has a proper structure that takes fewer data points to train and generalizes better to new queries. In [22, 23], neural operators are designed for solving partial differential equations and dynamical systems. They consist of integral operators and nonlinear activation functions so that their representation capability is general enough to approximate complex nonlinear operators.

As shown in Figure 1, the neural operator [23] composes linear integral operator $\mathcal{K}$ with pointwise non-linear activation function $\sigma$ to approximate highly non-linear operators.

where $\mathcal{P}:\mathbb{R}^{d_{a}}\to\mathbb{R}^{d_{1}}$, $\mathcal{Q}:\mathbb{R}^{d_{L}}\to\mathbb{R}^{d_{u}}$ are the pointwise neural networks that encode the lower dimension function into higher dimensional space and vice versa. The model stack $L$ layers of $\sigma(W_{l}+\mathcal{K}_{l})$ where $W_{l}\in\mathbb{R}^{d_{{l+1}}\times d_{l}}$ are pointwise linear operators (matrices), $\mathcal{K}_{l}:\{D\to\mathbb{R}^{d_{l}}\}\to\{D\to\mathbb{R}^{d_{l+1}}\}$ are integral kernel operators. We use ReLU activation in all numerical experiments, where $\sigma(x)=max(0,x)$ element-wise. The parameters $\theta$ of the neural operator to be learned consists of all the parameters in $\mathcal{P},\mathcal{Q},\{W_{l},\mathcal{K}_{l}\}_{l=1}^{L}$.

In the Fourier neural operator (FNO) [23], the integral kernel operators are restricted to convolution, and implemented with Fourier transform:

where $\mathcal{F}$ denotes the Fourier transform of a function $f:D\rightarrow\mathbb{R}^{d_{v}}$, $\mathcal{F}^{-1}$ denotes its inverse, and $K$ denotes the linear transformation matrix applied to function $(\mathcal{F}v_{t})$ in Fourier domain. An illustration is given in Figure 1.

Leveraging FNO’s powerful representation ability and superior computation efficiency, we propose to use it for learning and accelerating PDE observer computations. As we show here through our experiment results, FNO achieves up to three orders of magnitude improvement in computational speed compared to traditional PDE solvers, making FNO-PDE observers suitable for real-time control and monitoring.

In this section, we describe the proposed ML-accelerated PDE backstepping observers. We start with introducing the structure of a Learning-type PDE observer; we then present two model structures for implementation.

To illustrate the capability of the proposed ML-accelerated PDE observers, we compare the proposed observers with the conventional PDE observer on a chemical tubular reactor example (nonlinear parabolic PDEs) with both an exponential convergent backstepping observer and a prescribed-time convergent backstepping observer, and a backstepping traffic observer (nonlinear hyperbolic PDEs).

In order to demonstrate the effectiveness of FNO-backstepping pairing, we add a baseline long short-term memory (LSTM) model, which is a commonly used deep learning model capable of learning dependence in sequence prediction. Unless otherwise specified, we use $1000$ training instances and 100 testing instances. We use Adam optimizer to train for 500 epochs²²2Number of epochs defines the number times that the learning algorithm will work through the entire training dataset. with an initial learning rate of $0.001$ that is halved every 100 epochs. Both the recurrent neural operator and LSTM are trained with the teacher-forcing [24] that takes ground truth values as supervision during training.

We proposed an ML-accelerated PDE observer design framework which leverages the PDE backstepping design of observer gains and FNO for model learning. It highlights the benefits of inheriting the mathematical rigor of the PDE backstepping observers and obtain significant computational acceleration. We demonstrate the performance of the ML-accelerated observers in three benchmark PDE examples. There are many interesting open questions that remain. One important direction is to combine the ML-accelerated observers with downstream control tasks and study their real-time performance, especially in the presence of unmodeled PDE system disturbances.