Bridging Low- and High-Fidelity Simulations of Parametrized Time-Dependent PDEs with Multi-Fidelity Neural Operators
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Historically, the use of complex mathematical models in science and engineering has always implied severe computational issues, giving rise to a wealth of surrogate and reduced-order modeling strategies to tackle repeated simulations in a rapid and reliable way. In this work, we propose leveraging low-fidelity simulations, which are computationally inexpensive and physically structured, as inputs for neural operators designed to reconstruct high-fidelity solutions. Overall, we show how to provide a flexible multi-fidelity framework, where low-fidelity models may differ from high-fidelity counterparts either in spatial and temporal resolution or in the underlying physical modeling, just to mention two relevant cases still not addressed by well-established multi-fidelity reduced-order surrogate models. In this framework, neural operators learn mappings between infinite-dimensional function spaces, making them particularly well suited for modeling complex, time-dependent physical systems. Unlike conventional neural networks, they are independent of the underlying discretization and can capture long-range correlations, offering a mathematically principled framework that generalizes effectively and supports zero-shot super-resolution in both space and time. Among neural operators, we adopt a tensorized formulation derived from the original Fourier Neural Operator (TFNO), as proposed in (Kossaifi et al., 2023). This approach retains the spectral representation of the original FNO while introducing a parameter-efficient architecture with particular emphasis on accurate spectrum reconstruction. This spectral formulation preserves classical numerical concepts, such as spectral methods, while maintaining a strong connection to the underlying physics. Using low-fidelity simulations as inputs is especially advantageous for temporal problems, as they provide a shared representation and guide the system’s temporal evolution, offering a reliable and inexpensive history. The proposed framework demonstrates effectiveness across multiple test cases with varying physical models and fidelities, among which real-world structural mechanics problems driven by geometrical nonlinearities, typical of MEMS applications, are addressed.
