Approximate Feedback Linearization for a Nonlinear Hyperbolic PDE Class -- Part II: Neural Operator

Technical Summary: Approximate Feedback Linearization for a Nonlinear Hyperbolic PDE Class — Part II: Neural Operator

Problem Statement
This paper addresses the computational intractability of implementing exact feedback linearization for a class of nonlinear hyperbolic partial differential equations (PDEs). While Part I of this work established that a finite-order truncation of the Volterra series feedback linearizer provides closed-loop stability, the resulting controller remains computationally prohibitive for real-time application. Specifically, evaluating the truncated controller requires:

1. Solving a "tower" of $N-1$ plant-specific linear kernel PDEs on growing simplices to generate the feedback kernels.
2. Performing $N$-fold nested integrals over these precomputed kernels for every control update.

These operations are "plant-specific," meaning that any change in the plant's nonlinearity $F$ necessitates solving the kernel PDEs from scratch. This paper proposes replacing the explicit, computationally heavy truncated controller with a learned neural operator surrogate to eliminate online kernel solving and nested integration while preserving stability guarantees.

Methodology
The core methodology involves learning a mapping from the plant's nonlinearity coefficients and the system state directly to the boundary control input, bypassing the intermediate step of solving kernel PDEs.

1. Joint Lipschitz Continuity: The authors first establish that the truncated Volterra feedback operator, denoted $U_N(f_N, u)$, is jointly Lipschitz continuous with respect to both the plant coefficients $f_N$ (the Volterra kernels of the nonlinearity) and the state $u$. This is a critical theoretical step, as it ensures the operator is continuous on compact sets, satisfying the prerequisites for universal approximation.
2. Neural Operator Surrogate: Leveraging the Lipschitz property, the paper employs a neural operator (specifically a MIONet architecture) to approximate $U_N$. The network is trained on a dataset comprising pairs of plant coefficients and states mapped to the exact truncated control values. Once trained, the neural operator $\hat{U}_N$ acts as a direct surrogate, taking $(f_N, u)$ as input and outputting the control $U(t)$ in a single forward pass.
3. Closed-Loop Analysis: The stability of the closed-loop system under this surrogate feedback is analyzed. The boundary residual driving the target transport equation is shown to consist of two components: the truncation error (from Part I) and the neural operator approximation error ($\epsilon$).

Key Contributions
The paper makes three primary theoretical and practical contributions:

• Lipschitz Continuity Result: It proves that the truncated Volterra feedback operator is locally Lipschitz on bounded sets of plant coefficients and states. This result serves as the mathematical foundation for the subsequent approximation guarantees.
• Existence of Neural Surrogate: It establishes the existence of a neural operator approximation $\hat{U}_N$ capable of achieving any prescribed sup-norm accuracy on bounded sets of plants and states, effectively subsuming the kernel PDE solution and nested integration into a single learned mapping.
• Practical Stability Theorem: It proves that the closed-loop system under neural-operator feedback retains the four stability properties established in Part I (forward invariance, practical stability, finite-time attractivity, and class-$KL$ asymptotic estimates). Crucially, the asymptotic envelope is shown to be of the form $\|u(\cdot, t)\|_\infty \leq \beta(\|u_0\|_\infty, t) + \gamma(\epsilon)$, where the residual ball scales linearly with the training accuracy $\epsilon$.

Results
The paper validates the framework through numerical experiments at truncation order $N=3$:

• Stabilization: The learned controller successfully stabilizes nonlinear hyperbolic plants that are unstable in open loop.
• Robustness: The surrogate, trained on a specific distribution of plant parameters, stabilizes unseen plants (different nonlinearity parameters) and various initial conditions without retraining.
• Truncation vs. Approximation: The results demonstrate that while the neural operator introduces an approximation error $\epsilon$ (resulting in a residual floor of $\sim 10^{-3}$), it maintains stability even in regimes where lower-order truncations (e.g., $N=2$) fail due to large nonlinearity coefficients.
• Computational Efficiency: The most significant practical result is the computational speedup. The classical implementation of the $N=3$ controller requires solving kernel PDEs and nested integrals, taking approximately 3.95 seconds per evaluation. The trained neural operator evaluates the same control law in approximately 1 millisecond, yielding a speedup factor of roughly 4,000$\times$.

Significance and Claims
The paper positions itself as the practical realization of the theoretical framework introduced in Part I. While Part I made the controller "implementable in principle" by truncating the infinite series, this work makes it "implementable in practice, once and for all per plant class."

The authors claim that by learning the operator directly, the method eliminates the need for online kernel solving, which is the primary bottleneck for real-time control of nonlinear PDEs. The significance lies in the rigorous proof that this approximation does not sacrifice stability; the system remains practically stable with a residual error that is explicitly bounded and linearly proportional to the neural network's training accuracy. As the training accuracy approaches zero, the behavior of the exact truncated controller (and ultimately the exact infinite Volterra linearizer) is recovered.