On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations

1. Problem Statement

Context: Delay Differential Equations (DDEs) are ubiquitous in engineering and science (e.g., networked control, biological systems). Unlike Ordinary Differential Equations (ODEs), DDEs evolve on an infinite-dimensional phase space because the state at time $t$ depends on the history of the state over a delay interval $[-\tau_d, 0]$.

Challenges:

• Infinite Dimensionality: Standard data-driven methods like Extended Dynamic Mode Decomposition (EDMD) and Koopman operator theory are well-established for ODEs and Partial Differential Equations (PDEs) but struggle with DDEs due to the infinite-dimensional nature of the state (history functions).
• Existing Limitations: Current approaches to DDEs often rely on heuristic delay embeddings without quantifying approximation errors, or they suffer from high computational costs and training instability (e.g., adjoint-based neural DDEs).
• Gap: There is a lack of a theoretically grounded, data-driven framework for nonlinear DDEs that offers explicit, deterministic error guarantees for prediction and control.

2. Methodology

The authors propose a framework that bridges infinite-dimensional delay dynamics with finite-dimensional Koopman learning through history discretization and reconstruction.

A. Finite-Dimensional Realization

1. Discretization: The infinite-dimensional history function $\eta \in C([-\tau_d, 0], \mathbb{R}^n)$ is approximated by sampling it at $M$ uniform points $\Theta_M = \{\theta_j\}_{j=1}^M$. This maps the history to a finite-dimensional vector $z \in \mathbb{R}^{nM}$.
2. Operators:
   • Sampling Operator ($Q$): Maps the continuous history function to the discrete vector.
   • Reconstruction Operator ($R$): Maps the discrete vector back to a piecewise affine continuous function (linear interpolation between sample points).
3. Approximate Dynamics: The exact DDE flow $F$ is approximated on the discrete space by $\tilde{F} = Q \circ F \circ R$. This creates a closed-loop map on the finite-dimensional space $Z$, introducing a discretization error bounded by the grid spacing $\delta(M)$.

B. Data-Driven Linear Surrogate (kEDMD)

To learn the dynamics of $\tilde{F}$ without knowing the underlying physics:

1. Koopman Operator: The nonlinear map $\tilde{F}$ is lifted to a linear operator acting on observables (functions of the state).
2. Kernel-Based EDMD (kEDMD):
   • The authors use Wendland Radial Basis Function (RBF) kernels to define a Reproducing Kernel Hilbert Space (RKHS).
   • Instead of a global dictionary, they use a local regression approach. For each point in a dataset, they solve an affine regression problem using local neighbors to estimate the local dynamics.
   • These local estimates are aggregated to construct a global linear surrogate matrix $A$ that approximates the Koopman operator in the lifted space.
3. State Reconstruction: A mechanism is developed to recover the discretized state $z$ from the lifted Koopman coordinates using kernel interpolation, ensuring the prediction remains in the physical state space.

3. Key Contributions

The paper makes four primary theoretical and practical contributions:

1. Rigorous Koopman Realization for DDEs:
   • Establishes a formal bridge between the infinite-dimensional Banach space of history functions and a finite-dimensional Koopman representation via a sampling-reconstruction framework.
2. Data-Driven Linear Surrogate with kEDMD:
   • Constructs a finite-dimensional linear surrogate model using kEDMD with Wendland RBF kernels. This approximates the infinite-dimensional Koopman operator defined on the space of history functions.
3. Deterministic Error Guarantees:
   • Derives explicit error bounds for the learned predictor. The total error is decomposed into three interpretable components:
     • History Discretization Error: Scales with grid spacing $\delta(M)$.
     • Kernel Interpolation Error: Scales with the fill distance $h_X$ of the data points.
     • Data-Driven Regression Error: Scales with the local neighborhood radius $\rho$.
   • Provides bounds for both the lifted state evolution and the reconstructed physical state prediction.
4. Provable State Reconstruction:
   • Develops a kernel-based method to recover the discretized state from lifted coordinates, with provable bounds on the reconstruction error, ensuring reliability for simulation and control.

4. Numerical Results

The framework was validated on two nonlinear DDE systems:

1. Scalar DDE: A system with Hill-type nonlinearity.
2. Tumor-Immune Interaction Model: A 2D system with delay coupling.

Findings:

• Convergence: The prediction error decreases as the discretization resolution ($M$) increases and as the data density (fill distance $h_X$) improves.
• Accuracy: The method successfully reconstructed both the discretized history states and the current state values ($x(t)$) with high accuracy.
• Comparison: The model outperformed lower-resolution discretizations and demonstrated robustness across different test trajectories not seen during training.
• Visualization: Plots showed close agreement between true trajectories and predicted trajectories for both scalar and planar systems.

5. Significance and Impact

• Theoretical Rigor: This work moves beyond heuristic applications of Koopman theory to DDEs by providing provable, deterministic error bounds. This is crucial for safety-critical applications (e.g., medical control, aerospace) where "black-box" predictions are insufficient.
• Control Applicability: By providing a linear surrogate with known error bounds, the framework enables the design of model-based controllers for complex delay systems, which was previously difficult due to the infinite-dimensional nature of DDEs.
• Scalability: The use of local regression and kernel methods offers a tractable alternative to neural networks for delay systems, avoiding the training instability associated with backpropagation through time in delayed trajectories.
• Future Directions: The authors suggest this foundation can be extended to closed-loop control synthesis and high-dimensional networked systems with delays.

In summary, this paper provides a mathematically sound, data-driven methodology to approximate, predict, and potentially control nonlinear delay differential equations by transforming them into a finite-dimensional linear problem with quantifiable accuracy.