Time-Conditioned UNet Operator

• The paper introduces a split Koopman autoencoder where temporal conditioning is realized via a linear operator acting on latent states for efficient multi-step prediction.
• The model employs end-to-end autoencoding with recursive application of the Koopman operator to enforce latent-space linearization and robust state reconstruction over noisy channels.
• Empirical tests on a 4D inverted cart-pole system show that increased latent dimensions and transmit power reduce RMSE, validating the approach for distributed wireless monitoring.

A time-conditioned UNet operator is not a concept introduced or discussed in (Girgis et al., 2021), "Split Learning Meets Koopman Theory for Wireless Remote Monitoring and Prediction." The following entry provides a comprehensive summary—anchored strictly to the cited work—on the split Koopman autoencoder, which can be conceptualized as an operator-centric architecture trained with temporal conditioning via autoencoder lifting, latent-space linearization, and split deployment over wireless for prediction and remote observation.

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A time-conditioned UNet operator, in the context of (Girgis et al., 2021), refers to a neural operator architecture in which “conditioning on time” is effected by a finite-dimensional, linear operator acting on latent variables extracted from high-dimensional state sequences via an autoencoder. The system couples end-to-end autoencoding with a learned Koopman operator, and is split for use in a distributed setting. The conditioning is implicit in the formulation of the latent evolution and its deployment for multi-step temporal prediction over a noisy channel. Below, key aspects of this mechanism are systematically detailed.

1. Koopman-Theoretic Autoencoding with Temporal Conditioning

The central mathematical structure is

• A nonlinear state sequence $x_t \in \mathbb{R}^d$ generated by unknown dynamics $x_{t+1} = f(x_t)$,
• Mapped to a lower-dimensional latent via an encoder $\varphi: \mathbb{R}^d \to \mathbb{R}^n$, $z_t = \varphi(x_t)$,
• Advanced in time linearly: $z_{t+1} \approx K z_t$, where $K \in \mathbb{R}^{n \times n}$,
• Reconstructed to observation space by a decoder $\psi: \mathbb{R}^n \to \mathbb{R}^d$, $\hat{x}_t = \psi(z_t)$.

The term "time-conditioned" refers to the operator $K$ which governs latent evolution across time steps, enabling long-range predictions via repeated application: $z_{t+\tau} = K^\tau z_t$.

2. Split-Learning Architecture and Distributed Conditioning

A defining feature is "splitting" the autoencoder across a communication channel:

• The encoder $x_{t+1} = f(x_t)$0 is resident on a local sensor/edge device,
• Only the latent $x_{t+1} = f(x_t)$1 is transmitted over the channel (dimension $x_{t+1} = f(x_t)$2), resulting in substantial payload reduction,
• The receiver/observer hosts $x_{t+1} = f(x_t)$3 and $x_{t+1} = f(x_t)$4, and performs both reconstruction and prediction:
  • Decoding: $x_{t+1} = f(x_t)$5,
  • Temporal prediction: propagate $x_{t+1} = f(x_t)$6 using $x_{t+1} = f(x_t)$7 to obtain future $x_{t+1} = f(x_t)$8, decode to $x_{t+1} = f(x_t)$9.

After training, a "prediction-only phase" enables the observer to roll out future states using $\varphi: \mathbb{R}^d \to \mathbb{R}^n$0 and $\varphi: \mathbb{R}^d \to \mathbb{R}^n$1, without fresh measurements.

3. Joint Training Objective: Losses and Time-Linearization

Training minimizes a combined loss, balancing state reconstruction and temporal linearity in latent space:

$\varphi: \mathbb{R}^d \to \mathbb{R}^n$2

• The first term enforces faithful encoding/decoding (autoencoder fidelity),
• The second enforces that the latent sequence is nearly invariant under the linear operator $\varphi: \mathbb{R}^d \to \mathbb{R}^n$3 (i.e., time-conditioning in the observable space),
• Hyperparameter $\varphi: \mathbb{R}^d \to \mathbb{R}^n$4 regulates the tradeoff: large $\varphi: \mathbb{R}^d \to \mathbb{R}^n$5 induces stricter time-linearization, small $\varphi: \mathbb{R}^d \to \mathbb{R}^n$6 prioritizes reconstruction.

4. Implementation: Layer Profiling, Training and Inference Flow

Both encoder and decoder are fully-connected (FC) neural networks:

• Encoder: $\varphi: \mathbb{R}^d \to \mathbb{R}^n$7 with ReLU activations,
• Decoder: mirror structure, ReLU except final layer.

Training is performed using Adam optimizer ($\varphi: \mathbb{R}^d \to \mathbb{R}^n$8), batch size 128, and includes the impact of a Rayleigh fading channel and AWGN for realism.

At inference:

• The observer side computes $\varphi: \mathbb{R}^d \to \mathbb{R}^n$9 recursively, and predicts as many future steps as desired, decoding each via $z_t = \varphi(x_t)$0.

5. Empirical Behavior: Impact of Latent Dimension and Channel SNR

The architecture was tested on a 4D inverted cart-pole system. Results as a function of embedding dimension $z_t = \varphi(x_t)$1 and transmit power $z_t = \varphi(x_t)$2 reveal that:

• Increasing $z_t = \varphi(x_t)$3 (i.e., more Koopman modes) systematically reduces the RMSE of predicted states:
  • $z_t = \varphi(x_t)$4: RMSE $z_t = \varphi(x_t)$5 dBm (with $z_t = \varphi(x_t)$6W)
  • $z_t = \varphi(x_t)$7: RMSE $z_t = \varphi(x_t)$8 dBm (with $z_t = \varphi(x_t)$9W)
• Increasing transmit power $z_{t+1} \approx K z_t$0 reduces channel noise, improving observability and training convergence, also lowering RMSE.

This empirically validates the sensitivity of operator-based, time-conditioned architectures to both embedding size and communications quality.

6. Conceptual Extensions and System-Level Implications

The split Koopman autoencoder, as a canonical “time-conditioned operator”,

• Enables direct on-observer latent linear rollouts for temporal prediction with no further measurements,
• Provides a tractable route for integrating such predictions with downstream control (e.g., model-predictive control, possibly closing the loop by integrating predicted $z_{t+1} \approx K z_t$1 into controller logic),
• Generalizes to multi-sensor scheduling, spatiotemporal systems (with convolutional $z_{t+1} \approx K z_t$2), and co-design of resource-constrained inference under varying system and channel constraints.

7. Comparison with Broader Koopman and Autoencoder Literature

The structural approach here predates but is compatible with developments across the Koopman autoencoder literature:

• It enforces time-conditioning (latently linear temporal propagation) via a single global $z_{t+1} \approx K z_t$3,
• Contrasts with invertible designs (Tayal et al., 2023, Frion et al., 17 Mar 2025) and explicit forward-backward bias reduction (Singh et al., 5 Jan 2026),
• The use of split learning and wireless deployment positions it for resource-constrained, distributed deployment scenarios.

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In summary, the split Koopman autoencoder of (Girgis et al., 2021) instantiates a class of time-conditioned neural operators in which time invariance is enforced via linear evolution in autoencoder-lifted latent space, jointly trained for reconstruction and temporal linearization, and deployed split across sensor-observer boundaries for communication-efficient, long-horizon prediction in wireless remote monitoring. Conditioning on time is realized in the operator $z_{t+1} \approx K z_t$4, and exploited by recursive linear propagation and decoding at the observer.