Technical Summary: Learning Control-Affine Reduced-Order Models via Autoencoders

Problem Formulation

The paper addresses the challenge of identifying reduced-order models (ROMs) for high-dimensional, nonlinear dynamical systems governed by partial differential equations (PDEs) or complex simulations. Traditional projection-based methods (e.g., POD-Galerkin) require explicit knowledge of governing equations, which is often unavailable in data-driven scenarios. While deep learning approaches using autoencoders (AEs) have successfully identified low-dimensional latent manifolds, the resulting latent dynamics are often unstructured, complex, or purely linear.

A specific limitation arises when these models are intended for control applications. Purely linear latent dynamics may fail to capture complex nonlinear behaviors, while general nonlinear latent dynamics often lack the structural properties required for efficient control design (e.g., feedback linearization). The authors identify a gap in existing data-driven ROMs: the lack of a framework that enforces a control-affine structure on the latent dynamics, which balances nonlinear modeling capacity with the analytical tractability required for control.

Methodology

1. Control-Affine Latent Dynamics

The core contribution is a framework that learns a latent representation $z_k \in \mathbb{R}^r$ where the dynamics follow a discrete-time control-affine structure:
$$z_{k+1} = a_{\phi_1}(z_k) + B_{\phi_2}(z_k) \cdot u_k$$
Here, $a_{\phi_1}$ is a "drift" network representing autonomous dynamics, and $B_{\phi_2}$ is an "input" network that interacts with the control input $u_k$ in an affine manner. This structure is chosen because it aligns with many physical systems (e.g., robotics, chemical reactors) and enables the application of feedback linearization.

2. Joint Training Strategy

Unlike standard approaches that decouple autoencoder training from dynamics learning, this method employs a simultaneous training strategy. The encoder ($E_\theta$), decoder ($D_\phi$), and the latent dynamics model ($G_\phi$) are optimized together using a multi-objective loss function:
$$L = \gamma_1 L_{AE} + \gamma_2 L_{lat-cons} + \gamma_3 L_{dyn}$$

• $L_{AE}$: Reconstruction error of the high-dimensional state.
• $L_{lat-cons}$: Consistency loss ensuring the latent dynamics predict the encoded future states ($E_\theta(x_{k+l})$) accurately. This term is crucial for ensuring the latent space dynamics are consistent with the ground truth, preventing the decoder from compensating for poor latent dynamics.
• $L_{dyn}$: End-to-end error in the high-dimensional space.

The framework also supports input autoencoders to compress high-dimensional control inputs (e.g., distributed boundary fields) into a latent input vector $u'_k$, allowing the control-affine structure to handle high-dimensional actuation.

3. Sequence-Based Formulation

To improve prediction accuracy without sacrificing the control-affine structure, the authors extend the model to operate on sequences. An extended state vector $\xi_k$ is constructed containing histories of latent states and inputs. The dynamics are then modeled as:
$$\xi_{k+1} = \alpha(\xi_k) + \beta(\xi_k) \cdot u_k$$
where $\alpha$ includes a linear shift operator for memory and a nonlinear drift term, and $\beta$ applies the control input. This formulation allows the model to capture time-delay effects and history dependence while maintaining the control-affine property.

4. Feedback Linearization

The paper provides guidelines for applying feedback linearization to the learned ROM.

• Full-State Linearization: For the standard model, if the input matrix $B_{\phi_2}(z_k)$ has full row rank ($r \leq m$), a feedback law can be derived to transform the nonlinear dynamics into a linear system $z_{k+1} = \bar{A}z_k + \bar{B}v_k$.
• Input-Output Linearization: For the sequence-based model, full-state linearization is structurally impossible due to the memory buffer. However, the authors prove that exact input-output linearization is achievable by treating the current latent state as the output, provided the input matrix of the extended system has full rank. This allows the use of standard linear controllers (e.g., PID, LQR) in the latent space.

Key Contributions

1. Control-Affine ROM Framework: A novel data-driven method that enforces a control-affine structure on latent dynamics, bridging the gap between expressive nonlinear modeling and control-theoretic tractability.
2. Joint Optimization: A training strategy that simultaneously learns the coordinate transformation (AE) and the structured dynamics, ensuring the latent space is compatible with the control-affine assumption.
3. Sequence-Based Extension: An adaptation of the control-affine model to handle state and input histories, improving prediction accuracy while preserving the affine structure.
4. Theoretical Guarantees for Control: Formal propositions establishing conditions under which the learned ROMs can be exactly linearized via feedback, enabling the use of linear control techniques on nonlinear reduced-order models.
5. Open-Source Implementation: The authors release DeepE2EROM, a Python package implementing the end-to-end training framework.

Experimental Results

The framework was evaluated on two numerical examples:

1. One-Dimensional Heat Equation

• Setup: A distributed control problem where the state is a temperature field and the input is a spatially distributed heat source. Both state and input were high-dimensional.
• Performance: The control-affine ROM significantly outperformed a baseline ROM with linear latent dynamics. It achieved a two-fold reduction in end-to-end RMSE and a more pronounced improvement in latent dynamics accuracy.
• Control: The model was successfully used to track a reference temperature trajectory using feedback linearization combined with a PID controller. The controller successfully generated the necessary distributed heat inputs to steer the system.

2. Boxed Ball (2D Simulation)

• Setup: A ball moving in a unit box with strong nonlinear repulsive forces near boundaries, observed via 64x64 camera frames.
• Performance: The control-affine ROM outperformed the linear baseline in both forced (actuated) and autonomous settings. In the forced setting, it achieved roughly half the end-to-end RMSE and one-third the latent error of the baseline.
• Autonomous Dynamics: Even without explicit training on autonomous trajectories, the model accurately identified the system's equilibrium point.
• Control: The model enabled reference tracking of a circular trajectory using a proportional controller on the linearized latent dynamics.

Significance and Claims

The paper claims that enforcing a control-affine structure on latent dynamics offers a "powerful compromise." It retains the capacity for accurate nonlinear modeling while enabling the use of well-established linear control techniques (via feedback linearization) after a suitable coordinate transformation.

The authors emphasize that while linear latent dynamics are popular (often linked to Koopman operator theory), they may be insufficient for complex nonlinear systems. Conversely, general nonlinear latent models often lack interpretability and control tractability. The proposed control-affine approach addresses these limitations by naturally aligning with a wide class of engineering systems where dynamics are nonlinear in the state but affine in the control input.

The work is modest in its scope, noting that full-state feedback linearization requires the latent dimension to be less than or equal to the number of inputs ($r \leq m$), which can be restrictive for systems with few actuators. In such cases, the authors acknowledge that only partial feedback linearization is possible. Future work is suggested to address this by decomposing the latent space into controllable and unactuated subspaces.

The paper concludes that the proposed framework, validated on both PDE-based and image-based systems, provides a robust foundation for designing controllers for high-dimensional nonlinear systems directly from data.