Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs

Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles functional coefficients, dependent on the spatial variables, and gain functions dependent on space as well. The designs of gains for controllers and observers for PDEs, such as PDE backstepping, are mappings of system model functions into gain functions. These infinite dimensional nonlinear operators are given in an implicit form through PDEs, in spatial variables, which need to be solved to determine the gain function for each new functional coefficient of the PDE. The need for solving such PDEs can be eliminated by learning and approximating the said design mapping in the form of a neural operator. Learning the neural operator requires a sufficient number of prior solutions for the design PDEs, offline, as well as the training of the operator. In recent work, we developed the neural operators for PDE backstepping designs for first order hyperbolic PDEs. Here we extend this framework to the more complex class of parabolic PDEs. The key theoretical question is whether the controllers are still stabilizing, and whether the observers are still convergent, if they employ the approximate functional gains generated by the neural operator. We provide affirmative answers to these questions, namely, we prove stability in closed loop under gains produced by neural operators. We illustrate the theoretical results with numerical tests and publish our code on github. The neural operators are three orders of magnitude faster in generating gain functions than PDE solvers for such gain functions. This opens up the opportunity for the use of this neural operator methodology in adaptive control and in gain scheduling control for nonlinear PDEs.


Introduction
ML as a tool for learning PDE control methodologies.In Bhan, Shi, and Krstic (2023a) we introduced a learning-based control framework which devises a new role for machine learning (ML): learn an entire control design methodology, in the form of a mapping from the plant model to the controller gains, or even to the control inputs.Since the infinite-dimensional state of a PDE is a function of spatial variables, in PDE control the controller gain is also a function of spatial variables.Finding the gain typically entails solving a PDE in space (but not in time).It is therefore of interest, in PDE control, to have a capability where producing the control gain functions is just an evaluation of a neural mapping that has already learned the design methodology on a large set ✩ The material in this paper was partially presented at The 62nd IEEE Conference on Decision and Control (CDC), December 13-15, 2023, Marina Bay Sands, Singapore.This paper was recommended for publication in revised form by Associate Editor Thomas Meurer under the direction of Editor Luca Zaccarian. of previously offline-solved control design problems for a sample set of PDEs in a certain class.
Neural operators for approximating mappings of functions into functions.Our inspiration for encoding PDE control methodologies into machine learning comes from recent advances in the mathematics of machine learning.Motivated by the tasks of finding solution/flow maps (from the initial conditions into future states) for physical PDEs, several machine learning research teams (Li et al., 2021;Lu, Jin, Pang, Zhang, & Karniadakis, 2021;Seidman, Kissas, Perdikaris, & Pappas, 2022) have developed neural approximation methods, termed ''neural operators''.These approaches are not simply discretizing PDEs and finding solution maps to the resulting large ODE solution problems.They approximate (nondiscretized) function-to-function nonlinear operators and provide provable guarantees of the accuracy of approximation in terms of the required sizes of the training sets and neural networks.Neural operators have demonstrated impressive computational speedups in solving complex PDE systems for weather forecasting (Kurth et al., 2023), state estimation and control (Bhan, Shi, & Krstic, 2023b;Shi et al., 2022), and producing numerical solutions to previously unsolvable problems such as high-dimensional PDEs (Han, Jentzen, & Weinan, 2018).
With a rigorous and numerically powerful capability like this, specific PDE control methods, for specific classes of PDEs, can be learned once and encoded as neural operators, ready to produce the control gain functions for any new functional coefficients of the same classes of PDEs.
Such an approach is of value if it allows the retention of the theoretical properties proven for the ''exact design''.This is indeed what we show in the paper (Bhan et al., 2023a) in which we introduce the framework: approximate neural operator representations of a particular PDE control method -PDE backstepping -preserves its stability guarantees in spite of the control gains not being generated by solving the design PDEs but by the gains being generated from the learned ''neural model'' of PDE backstepping.
Extension of PDE backstepping neural operators from hyperbolic (Bhan et al., 2023a) to parabolic PDEs.Hyperbolic PDEs involve only the first derivatives in space and time.This makes them (all else being equal) the ''simplest'' PDE class for control.Delay systems combine ODEs with delays-the simplest form of a PDE.While the simplest among PDEs, hyperbolic PDEs are not necessarily easy to control.They can be unstable, with many unstable eigenvalues, and only one input acting at the boundary of a domain.This mix of simplicity within the PDE family, with the non-triviality for control, makes hyperbolic PDEs the ideal entry point for any new study in PDE control, including the introduction of a new framework for learning-based control in our (Bhan et al., 2023a).
Control design problems for hyperbolic PDEs are hyperbolic PDEs themselves, namely, equations with only first derivatives in multiple spatial variables.Parabolic PDEs, with their first derivative in time but second derivatives in space, are the natural next challenge for learning the PDE backstepping methodology using neural operators.This is what we undertake in this paper.The chief difficulty with learning backstepping kernel operators for parabolic PDEs is that the kernels are governed by second-order PDEs, which raises the difficulty for solving such PDEs and for proving the sufficient smoothness of their solutions so that the neural operator (NO) approximations have guarantee of sufficient accuracy for preserving stabilization.
We consider parabolic PDE systems of the form ( Our goal is the design of a PDE backstepping boundary control as well as an observer with the (collocated) boundary sensing of u x (1, t).By ''design'' we mean to find the gain function k in the control law (4), namely, to find the output k of the function-tofunction mapping K : λ ↦ → k, depicted in Fig. 1.This paper's objective is to learn the design operator K with a neural operator approximation K and to employ the resulting approximate gain k in the control law.
Since parabolic PDEs in one dimension have two boundary conditions, and also boundary actuation and boundary sensing can be employed at either boundary, a total of twelve combinations of boundary actuation, boundary sensing, and boundary condition on the unactuated boundary are possible.Taking the symmetry between the boundaries x = 0 and x = 1 into account, the total number of truly distinct combinations is six.They are listed in Table 1.
We are able to solve all six problems but, in this paper, pursue the simplest of the six combinations for pedagogical reasons.The case with Dirichlet boundary conditions, u(0, t) = 0, u(1, t) = U(t) is, notationally, the simplest case.It allows the reader to most quickly grasp the utility and the technical steps in employing neural operators in the control of parabolic PDEs.We opt to present in the paper the results for the combination in the second row of Table 1 because this combination allows us to ''kill two birds with one stone'' in our exposition.For this particular actuator-sensor combination, which is collocated (and the simplest of the four collocated combinations), the same kernel is used to obtain the gain functions for both the controller and the observer.Further, using the blueprint presented here, readers interested in the other combinations in Table 1 can develop controllers and observers for any sensing and control application.
All the results in the paper -a full-state controller, an observer, and an output-feedback law (as well as the seven additional combinations not pursued in the paper) -can be extended to the more general class of parabolic PDE systems, (5) PDE backstepping.Even though PDE backstepping was first developed for parabolic systems (Smyshlyaev & Krstic, 2004), it is best to begin its study from the easier, hyperbolic case (Krstic & Smyshlyaev, 2008c).Control of hyperbolic PDEs has grown into a rich area because, in the hyperbolic case, one can stabilize a coupled system with fewer inputs than PDEs.A pair of coupled hyperbolic PDEs was stabilized with a single boundary input in Coron, Vazquez, Krstic, and Bastin (2013), an extension to n + 1 hyperbolic PDEs with a single input was introduced in Di Meglio, Vazquez, and Krstic (2013), an extension to n + m PDEs with boundary actuation on m ''homodirectional'' PDEs in Hu, Di Meglio, Vazquez, andKrstic (2016), Hu, Vazquez, Di Meglio, andKrstic (2019), an extension to cascades with ODEs in Di Meglio, Argomedo, Hu, and Krstic (2018), and an extension to ''sandwiched'' ODE-PDE-ODE systems in Wang andKrstic (2020, 2022).Redesigns robust to delays are provided in Auriol, Bribiesca-Argomedo, Saba, Di Loreto, and Di Meglio (2018).PDE backstepping-based output-feedback regulation with disturbances is proposed in Deutscher (2015), Deutscher and Gabriel (2018).
From an implementation perspective, approximation of the kernel PDEs has been explored in Woittennek, Riesmeier, and Ecklebe (2017) and in toolboxes such as Fischer, Gabriel, and Kerschbaum (2022).Furthermore, for observer problems, Grüne and Meurer (2022) decouples the infinite dimensional PDE system into a finite dimensional slow and infinite dimensional fast subsystem for controller implementation.However, a key difference is that this work learns a kernel, which can be of high value in adaptive control when the kernel PDE needs to be resolved repeatedly (or numerically approximated).
DeepONet.Using the basic notions and notation for NOs given in Bhan et al. (2023a), we state next the key technical result that enables our use of NOs to learn the PDE backstepping kernel mappings.The result is quoted in its general/abstract form.It is specialized to the PDE control setting in our Theorem 4.

Basic backstepping for reaction-diffusion PDE
We employ the following backstepping transformation, to convert (1), ( 2), (3) into the target system w(0, t) = 0 , w(1, t) = 0 , with the help of feedback (4).We could as well pursue the target system w t = w xx −cw, c > 0, but we forego this design flexibility for the sake of simplicity.
Theorem 4. For all B λ , B λ ′ > 0 and ϵ > 0, there exists a neural operator M such that, for all (x, y) ∈ T , holds for all Lipschitz λ with the properties that ∥λ∥ ∞ ≤ B λ , ∥λ ′ ∥ ∞ ≤ B λ ′ , namely, there exists a neural operator K such that K(λ)(x, 0) ≡ 0 and Note, we refer to the error ϵ between the operator and the neural operator as the approximation accuracy throughout the remainder of this paper.

Stabilization under DeepONet gain feedback
The following theorem establishes the properties of the feedback system with the approximated kernel.
Theorem 5. Let B λ , B λ ′ > 0 be arbitrarily large and consider the system (1), ( 2), (3) with any λ ∈ C 1 ([0, 1]) whose derivative λ ′ is Lipschitz and which satisfies ∥λ∥ ∞ ≤ B λ and ∥λ ′ ∥ ∞ ≤ B λ ′ .There exists a sufficiently small ϵ * (B λ , B λ ′ ) > 0 such that the feedback law with all NO gain kernels k = K(λ) of approximation accuracy ϵ ∈ (0, ϵ * ) in relation to the exact backstepping kernel k = K(λ) ensures that the closed-loop system satisfies the exponential stability bound where Proof.Approximate backstepping transform and perturbed target system.Take the backstepping transformation where k = K(λ).With the control law ( 22), the target system becomes ŵt (x, t) = ŵxx (x, t) + δ k0 (x)u(x, t) where With ( 21), we get Inverse approximate backstepping transformation.Since the state u appears under the integral in the ŵ-system (26), in the Lyapunov analysis we need the inverse backstepping transformation It is shown in Krstic and Smyshlyaev (2008a) that the direct and inverse backstepping kernels satisfy in general the relationship The inverse kernel satisfies the following conservative bound Since ∥k − k∥ ∞ < ϵ, we have that ∥ k∥ ∞ ≤ ∥k∥ ∞ + ϵ.With ( 14) we get and hence Lyapunov analysis.The Lyapunov functional has a derivative where With several straightforward majorizations, we get and  51), ( 52), ( 53) uses boundary measurement of the flux u x (1, t).The gain k(1, t) is produced with the DeepONet K.

Observer design
State estimators (observers) with boundary measurements can be formulated with four measurement choices on the interval [0, 1]: the measured quantities can be u(0, t), u x (0, t), u(1, t), u x (1, t).That leads to many possible problem formulations.Since our goals with observers are twofold -to estimate the unmeasured state but also to use it in output-feedback control for stabilization -our choice of measurement needs to be consistent with the actuation choice we have already pursued in this note, namely, Dirichlet actuation of u(1, t) = U(t).So, we cannot use u(1, t) for measurement but we can use u(0, t), u x (0, t), u x (1, t).We let the output u x (1, t) be measured.Our choice of u x (1, t) for measurement, as indicated in the observer diagram in Fig. 2, is motivated by the pedagogical fact that, with this measurement, an observer can be built using the same kernel k(x, y) as for the control law.Theorem 6.Let B λ , B λ ′ > 0 be arbitrarily large and consider the system (1), ( 2), (3) with any λ ∈ C 1 ([0, 1]) whose derivative λ ′ is Lipschitz and which satisfies ∥λ∥ ∞ ≤ B λ and ∥λ ′ ∥ ∞ ≤ B λ ′ .There exists a sufficiently small ϵ * (B λ , B λ ′ ) > 0 such that the observer û(0, t) = 0, ( 52) with all NO gain kernels k = K(λ) of approximation accuracy ϵ ∈ (0, ϵ * ) in relation to the exact backstepping kernel k = K(λ) ensure that the observer error system, for all u 0 , û0 ∈ L 2 [0, 1], satisfies the exponential stability bound where M(ϵ, λ) is defined in (24).

Numerical results
In Fig. 3, we show that the system (1), ( 2), ( 3) is open-loop unstable for the reaction term λ(x) = 50 cos(γ cos −1 (x)) for γ = 5, 8.The increased oscillation in larger γ yields a lower rate of instability as shown on the right.We simulate the PDE and its control using the finite difference scheme in the Appendix.In Fig. 4, we demonstrate both the analytical and learned DeepONet kernels for the two γ values corresponding to Fig. 3. To learn the mapping K : λ(x) ↦ → k(x, y), we construct a dataset of 1000 different λ(x) as the Chebyshev polynomials defined in λ(x) above with γ ∼ uniform (4, 9) and a 90/10 train test split.We choose λ of this form due to the rich set of kernel functions generated by varying only a single parameter.
To effectively utilize the DeepONet without modifying the grid, we stack λ(x) repeatedly n y times over the y axis to make a 2D input to the network.Then, we capitalize on the 2D mapping by implementing a CNN for the DeepONet branch network.In the future, one can explore neural operators on irregular girds along the direction of Li, Huang, Liu, and Anandkumar (2022).
For training, the relative L 2 error is 3.5e − 2 and the testing error is 3.6e − 2. With the learned neural operator, we achieve speedups on the magnitude of 10 3 compared to an efficient finite difference implementation.In Fig. 5, we demonstrate closed-loop stability with the neural operator approximated gain function for the control feedback law.Additionally, we see the error is largest at the beginning achieving a maximum in both cases of approximately 10%.In Fig. 6, we test the observer (52), ( 52), ( 53) with a DeepONetapproximated kernel trained as above using λ(x) = 20 cos(5 cos −1 (x)) with γ ∼ uniform (4, 9).Additionally, we apply a boundary signal of U(t) = 7 sin(16π t) + 10 cos(2π t) to generate a challenging and rich PDE motion for estimation.The true system state begins with initial conditions u(x, 0) = 10 while the DeepONet observer has initial conditions of ûNO (x, 0) = 20.
Despite this, the observer approximates the PDE well with a peak error of less than 5% compared to the analytical observer while maintaining the same 10 3 x speedup over the finite difference scheme.Lastly, although we chose λ(x) to be the Chebyshev polynomials, we emphasize one could chose any set of functions in C 2 [0, 1] such as polynomials, random Fourier basis, Legendre polynomials, etc. as long as those functions create a sufficiently diverse set of kernels for the neural network to satisfy the ϵ * .For an exact size of neural network to satisfy ϵ * , one can utilize the conservative estimates in Deng et al. (2022) although, in many cases, they are practically infeasible due to the extremely difficult challenge of operator approximation.

Conclusions
We extend the NO-powered PDE backstepping introduced in Bhan et al. (2023a) from the hyperbolic to parabolic PDEs, which poses additional challenges in advancing from first-order to second-order Goursat PDEs for the kernels, and advancing from the Lyapunov study of hyperbolic target PDEs with relatively  The peak error between the analytical observer (not shown) and the neural operator observer is around 0.3.simple perturbations to parabolic target PDEs with more complex perturbations.Additionally, we present the first extension of DeepONet-backstepping from full-state feedback to observer design.

Fig. 1 .
Fig. 1.The PDE backstepping design operator K : λ ↦ → k, where λ(x) is the spatially-varying reaction coefficient of the PDE, whereas k(x, y) is the kernel function of the backstepping transformation, producing the feedback gain function k(1, y) in the feedback law U(t) = ∫ 1 0 k(1, y)u(y, t)dy.

Fig. 3 .
Fig. 3. Open-loop instability (top) for the two respective reaction coefficients λ(x) with γ = 5 on the left and γ = 8 on the right.The top row shows the open loop stability with the coefficient and bottom row shows the function λ(x) with corresponding γ value.

Fig. 4 .
Fig. 4. Examples of the kernel k(x, y) (left column), learned kernel k(x, y) (middle column), and the kernel error k(x, y) − k(x, y) (right column).The two respective λ(x) values correspond to the same respective values as in Fig. 3 with γ = 5 on the top row and γ = 8 on the bottom row respectively.

Fig. 5 .
Fig. 5.For the two respective λ(x) values as in Fig. 3, the top row showcases closed-loop solutions with the learned kernel k(x, y), whereas the bottom row shows the closed-loop PDE error between applying the original kernel k(x, y) and the learned kernel k(x, y).

Table 1
Six possible combinations of boundary actuation, sensing, and boundary condition at the opposite end of [0, 1].We focus on the simplest combination-in the second row.