RONOM: Reduced-Order Neural Operator Modeling
This work addresses computational bottlenecks in physics-based modeling for applications like real-time forecasting and optimal control, offering an incremental improvement by combining existing concepts with new error bounds.
The authors tackled the challenge of computational intensity in solving time-dependent partial differential equations for many-query scenarios by introducing the RONOM framework, which bridges reduced-order modeling and neural operator learning. The results show that RONOM achieves comparable input generalization and superior spatial super-resolution and discretization robustness compared to existing neural operators, with novel insights into temporal super-resolution.
Time-dependent partial differential equations are ubiquitous in physics-based modeling, but they remain computationally intensive in many-query scenarios, such as real-time forecasting, optimal control, and uncertainty quantification. Reduced-order modeling (ROM) addresses these challenges by constructing a low-dimensional surrogate model but relies on a fixed discretization, which limits flexibility across varying meshes during evaluation. Operator learning approaches, such as neural operators, offer an alternative by parameterizing mappings between infinite-dimensional function spaces, enabling adaptation to data across different resolutions. Whereas ROM provides rigorous numerical error estimates, neural operator learning largely focuses on discretization convergence and invariance without quantifying the error between the infinite-dimensional and the discretized operators. This work introduces the reduced-order neural operator modeling (RONOM) framework, which bridges concepts from ROM and operator learning. We establish a discretization error bound analogous to those in ROM, and get insights into RONOM's discretization convergence and discretization robustness. Moreover, two numerical examples are presented that compare RONOM to existing neural operators for solving partial differential equations. The results demonstrate that RONOM using standard vector-to-vector neural networks achieves comparable performance in input generalization and superior performance in both spatial super-resolution and discretization robustness, while also offering novel insights into temporal super-resolution scenarios.