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

Abstract: Volterra series feedback linearizes a class of nonlinear hyperbolic PDEs but produces a controller that, even after truncation, demands solving a tower of plant-specific kernel PDEs and evaluating nested integrals. We prove the truncated controller is jointly Lipschitz in plant and state, and learn it as a single neural operator from plant nonlinearity and state to boundary control. Once trained, no kernel is ever solved again, for any plant in the trained class. The closed loop is practically stable in class-$\mathcal{KL}$ form, with a residual ball scaling linearly with training accuracy.

• The paper establishes that a trained neural operator can universally replace truncated Volterra feedback, eliminating the need for complex kernel PDE solutions.
• It leverages DeepONet-based universal approximation to maintain closed-loop practical stability with predictable terminal residuals tied to network accuracy.
• The neural operator surrogate delivers over 4000× speedup in computation, enabling real-time, adaptive control across a range of nonlinear hyperbolic PDE systems.

Neural-Operator Surrogates for Feedback Linearization of Nonlinear Hyperbolic PDEs

Introduction and Motivation

This work addresses the longstanding bottleneck in the feedback linearization of nonlinear hyperbolic partial differential equations (PDEs) governed by Volterra series nonlinearities. Conventional Volterra feedback theoretically achieves exact linearization, transforming the nonlinear PDE into a transport equation. However, the resulting controller is an infinite series involving high-order kernel PDEs and nested integrals, precluding practical implementation due to combinatorial explosion in complexity. Even after truncation at a modest order $N$, realization of the controller entails expensive, plant-specific kernel computations and $N$-fold numerical integration—operations that must be recomputed whenever the plant nonlinearity varies, severely limiting real-time applicability and adaptability.

The present paper establishes that a neural operator can, once trained offline, universally replace the truncated Volterra feedback across a class of plant nonlinearities and states, eliminating both the need for kernel PDE solves and online numerical quadrature. Crucially, all the essential closed-loop guarantees—forward invariance, practical stability, finite-time attractivity, and class-$\mathcal{KL}$ bounds—are preserved up to a network-accuracy-determined residual. This neural surrogate allows nonlinearity-aware boundary control in essentially constant time per evaluation.

Operator-Theoretic Framework and Main Results

The system of interest is a one-dimensional hyperbolic PDE with Volterra-nonlinear source term: $u_t(x,t) = u_x(x,t) + F[u(\cdot, t)](x),$ with Dirichlet actuation $u(0,t)=0$, $u(1,t)=U(t)$ and $F$ defined as an infinite sum of multilinear Volterra operators parameterized by kernel coefficients. The feedback law, in its truncated form, comprises a finite sum up to order $N$, each term involving nested integration against plant-dependent kernels. Notably, both the computation of these kernels and their evaluation are nonlinear, high dimensional, and plant-specific.

Key contributions include:

• Joint Lipschitz Continuity: It is rigorously established that the truncated feedback operator $\mathcal{U}_N(\mathbf{f}_N, u)$ is jointly Lipschitz in the Volterra coefficients $\mathbf{f}_N$ and the state $N$0, on bounded, regular sets. This continuity underpins the universal approximation result for operator learning.
• Neural Operator Universality: Leveraging the DeepONet universal approximation theorem, the paper shows that for any prescribed sup-norm accuracy on admissible classes of plant coefficients and states, there exists a neural operator $N$1 uniformly approximating the truncated feedback. Once trained, this surrogate eliminates all dependence on the intermediate kernel PDEs.
• Closed-Loop Practical Stability: All stability guarantees established in the companion work for the truncated feedback persist when the neural operator surrogate is deployed in the loop. Explicitly, the state norm converges to a residual $N$2 ball whose radius is linear in the network's approximation error and the truncation tail, yielding the estimate:

$N$3

where $N$4 is linear in the neural operator's sup-norm training error $N$5, and $N$6 is class-$N$7; as $N$8 the ideal limit is recovered.

Methodology: Operator Learning Surrogate

In the proposed architecture, the cascade comprising plant-coefficient-to-kernel mapping, kernel integration, and feedback sum is collapsed into a single neural operator. Training data are generated by evaluating the classical feedback law on synthetic samples from bounded sets of plant parameters and states (as dictated by the closed-loop reachability domains). The input consists of function evaluations of the Volterra kernels and the PDE state; the output is the corresponding boundary control value.

The surrogate typically adopts the MIONet/Transformer hybridization, ingesting the function-valued inputs as token arrays, and is trained to minimize mean-squared error to the numerically-computed true feedback. The universality and robustness of the approach arise from the underlying regularity and compactness of the admissible function classes, as justified via detailed operator-theoretic analysis.

Closed-Loop Behavior and Practical Stability

Under the neural surrogate, the closed-loop system can be viewed as a perturbed transport equation where the boundary residual is the sum of truncation and neural-approximation errors. The analysis shows that:

• Persistent Residuals: The neural operator's uniform error manifests as a persistent boundary disturbance, resulting in the state norm plateauing at a level proportional to $N$9. This regime is formalized in the practical stability Theorem, leading to a class-$\mathcal{KL}$0-type convergence envelope.
• Explicit Bounds and Contraction Analysis: The critical spectral gap (small-gain) and basin of attraction estimates are inherited from the underlying Volterra theory, provided the neural operator is trained in regions where the feedback's Lipschitz constant remains sub-unitary.
• No Kernel Recomputations: For any plant within the trained admissible class, the neural operator obviates all online kernel computation. The cumulative effect, in practice, amounts to guaranteeing stabilization for all parameter variations encompassed by the training corpus, with a one-time model training cost.

Numerical Experiments and Quantitative Results

Closed-loop experiments validate the correctness, robustness, and efficiency of the neural operator controller:

• Stabilization Across Plant Variations: As visualized in the closed-loop trajectories, both the exact and neural-operator feedback achieve stabilization from large (unstable) initial conditions, with the neural surrogate incurring only a small, predictable terminal error.

(Figure 1)

Figure 1: Closed-loop comparison for a representative plant, highlighting rapid stabilization under both exact and neural-operator feedback, with the surrogate plateauing to a small residual error.

• Robustness to Unseen Plants and Initial Conditions: The neural operator, trained on only a subset of the plant space, generalizes to distinctly parameterized, previously-unseen plants and a wide set of initial conditions, consistently producing bounded, stable closed-loop signals.

(Figure 2)

Figure 2: Robustness of the neural operator across a sweep of plant coefficients and initial conditions, demonstrating stable performance without additional retraining.

• Impact of Truncation Order: As the truncation order increases, the neural operator continues to stabilize even for larger-amplitude nonlinearities where lower-order truncations fail (unstable). The empirical difference between complete third-order and only quadratic controllers becomes pronounced as the problem nonlinearity intensifies.

(Figure 3)

Figure 3: Truncation-order ablation study at fixed plant nonlinearity, revealing that higher-order feedback and its neural surrogate are necessary for robust control in the strong-nonlinearity regime.

• Computation Time: Runtime profiling indicates that the neural surrogate executes in approximately $\mathcal{KL}$1 millisecond per feedback evaluation—yielding a more than $\mathcal{KL}$2 speedup over the classical feedback computation pipeline, which may require seconds per invocation due to kernel assembly and quadrature.

Theoretical and Practical Implications

The results formalize and practically realize the decoupling of feedback law evaluation complexity from plant-specific kernel computation. In high-stakes, real-time, and adaptive control scenarios for nonlinear PDE systems (e.g., fluid flows, traffic, chemical reactors), the neural operator approach affords previously unattainable computational tractability and reusability of controllers across a parametric family of plants.

The theoretical framework generalizes beyond the specific Volterra/transport setting; similar operator-learning surrogates could be deployed in diverse feedback design pipelines, especially where control laws are synthesisable via universal approximation and satisfy locality and regularity properties in the relevant function spaces.

In the context of AI-driven scientific computing, this work exemplifies the integration of rigorous control-theoretic guarantees with modern operator learning techniques, enforcing correctness while achieving ML-level efficiency.

Conclusion

A neural operator surrogate, trained offline as a functional approximation to a truncated Volterra feedback controller, universally replaces intricate kernel-based feedback laws for a class of nonlinear hyperbolic PDEs. This eliminates the online cost of kernel PDE solutions and numerical integration, maintains all essential closed-loop properties, and enables orders-of-magnitude speedup. The methodology is robust to plant and state variations within the admissible class, with terminal state regulation determined by controllable neural approximation and truncation errors. The approach paves the way for real-time, adaptive, and broadly transferrable controller deployment in nonlinear distributed parameter systems.