RHYME-XT: A Neural Operator for Spatiotemporal Control Systems

Marijn Ruiter; Miguel Aguiar; Jake Rap; Karl H. Johansson; Amritam Das

cs.LGMar 18, 2026

RHYME-XT: A Neural Operator for Spatiotemporal Control Systems

Authors: Marijn Ruiter, Miguel Aguiar, Jake Rap, Karl H. Johansson, Amritam Das

Organizations: Digital Futures and Division of Decision and Control Systems, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden. · Control Systems Group, EE Dept., Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

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 & tables

Panel of Figure 1 — Fig. 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 $\hat{u}(\cdot,t)$ .

Fig. 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.

Fig. 3: Box plot of the relative

\ell^{2}

error for different time horizons

T

, with the model being trained on trajectories with

T=50

Fig. 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

Fig. 5: Training loss trajectory for different training scenarios.

PDE Surrogate Modeling

PDE surrogate modeling builds fast approximations of partial differential equation solutions for simulation, prediction, and design.

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Aug 25, 2026J/K move · Enter open · S save

Physics-Integrated Operator Learning via Gaussian Splatting Representations

Jihao Zhang, Junyi Guo, Jian-Xun Wang

Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA

PDE Operator Learning PDE Surrogate Modeling Physics-Informed Neural Operators

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Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors during long-horizon autoregressive prediction and may fail to exploit available governing-equation structure. Existing approaches incorporate physics primarily through residual-based training objectives or PDE-specific architectural constraints, which can introduce optimization difficulties or limit architectural generality. In this work, we introduce a representation-level approach to physics integration in which a feed-forward Gaussian splatting (FFGS) representation serves as a continuous interface between discretized solution fields and governing operators. The FFGS representation reconstructs the state as a continuous Gaussian field with closed-form spatial derivatives, allowing available physical PDE operators to be integrated directly within the learned evolution map without introducing a physics-residual loss. We evaluate the framework across two- and three-dimensional PDE systems, including advection, diffusion, nonlinear self-advection, and reaction dynamics. Over long-horizon autoregressive rollouts, the proposed framework reduces relative

\ell_2

error by

1.5\times

2.2\times

compared with the strongest purely data-driven baseline across the benchmark suite, while consistently improving spectral fidelity. The framework also remains effective when the governing equations are partially known, demonstrating robustness to incomplete physics. These results demonstrate that continuous field representations can provide a practical interface for incorporating known physical structure into generic neural-operator surrogates.

Abstract

PDE Surrogate Modeling

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