RHYME-XT: A Neural Operator for Spatiotemporal Control Systems

Abstract

We propose RHYME-XT, an operator-learning framework for surrogate modeling of spatiotemporal control systems governed by input-affine nonlinear partial integro-differential equations (PIDEs) with localized rhythmic behavior. RHYME-XT uses a Galerkin projection to approximate the infinite-dimensional PIDE on a learned finite-dimensional subspace with spatial basis functions parameterized by a neural network. This yields a projected system of ODEs driven by projected inputs. Instead of integrating this non-autonomous system, we directly learn its flow map using an architecture for learning flow functions, avoiding costly computations while obtaining a continuous-time and discretization-invariant representation. Experiments on a neural field PIDE show that RHYME-XT outperforms a state-of-the-art neural operator and is able to transfer knowledge effectively across models trained on different datasets, through a fine-tuning process.

Figures (5)

• Figure 1: RHYME-XT architecture. The model inputs ($u_0$ and $f$) are shown in Purple. Green denotes the basis function generation, parameterized by the FNN $h_{\phi}$, where its inputs first pass through a random Fourier mapping $\gamma$. Red denotes the flow function architecture, taking in the projected inputs to generate the temporal coefficients. Blue denotes the output reconstruction, parameterized by the FNN $h_u$, which combines the basis functions and temporal coefficient to give the approximated solution. Specifically, the inputs $u_0$ and $f$, are projected using the basis functions to obtain $\mathbf{a}_0$ and $\mathbf{f}(t)$, respectively. The projected initial condition is mapped to $z_0$ by the encoder $h_\mathrm{enc}$, after which it is propagated in time through the RNN with input parameters $\{\mathbf{f}_k, \tau_k\}_{k=0}^{k_t}$, with $k_t=3$ in this example. The approximated temporal coefficients $\hat{\mathbf{a}}(t)$ are obtained by passing the output RNN state $z$ through the decoder $h_\mathrm{dec}$. These temporal coefficients are combined with the basis functions in the output reconstruction network to obtain the approximated output $\hat{u}(\cdot,t)$.
• Figure 2: Visualization of a trajectory of our proposed method RYHME-XT versus the baseline DeepONet. The relative $\ell^2$ errors of the trajectory are $0.0571$ for RHYME-XT and $0.4810$ for DeepONet, respectively.
• Figure 3: Box plot of the relative $\ell^2$ error for different time horizons $T$, with the model being trained on trajectories with $T=50$.
• Figure 4: Visualization of a trajectory generated with $T=250$ versus model predictions of our proposed method RHYME-XT, trained using trajectories with $T=50$. Ground truth: (-), RHYME-XT: (--). The relative $\ell^2$ error of this trajectory is $0.1173$.
• Figure 5: Training loss trajectory for different training scenarios.