Benchmarking Multi-fidelity Neural Operators on Complex PDE Problems with Non-trivial Fidelity Differences

2026-08-05

AI Summary

High-fidelity partial differential equation (PDE) data are often scarce and computationally expensive, limiting the performance of neural operators. This study systematically evaluates multi-fidelity neural operator strategies across four canonical PDE problems, comparing two-step methods, residual-based approaches, intermediate fusion, and transfer learning. It introduces two novel test cases that simulate realistic model mismatches beyond mere grid resolution differences—such as the nontrivial fidelity gap between Reynolds-Averaged Navier–Stokes (RANS) and Large Eddy Simulation (LES). The findings reveal that naively fusing low-fidelity predictions can amplify errors, whereas transfer learning with pre-trained weights significantly enhances high-fidelity prediction accuracy under complex fidelity discrepancies, demonstrating superior robustness.

Abstract

Solving PDE-governed physical problems is computationally expensive, limiting the availability of high-fidelity (HF) data for training neural operators, which typically require large datasets. Multi-fidelity learning addresses this by combining abundant low-fidelity (LF) data with limited HF samples. To evaluate several multi-fidelity neural operator strategies, including two-step, residual, intermediate, and transfer learning, we conduct a systematic assessment across four PDE test cases: two standard grid-based problems, a modified Darcy flow with LF-HF governing equation discrepancies, and an unsteady smoke inflow with temporal dynamics. Most prior studies focus on grid-based LF-HF differences, but real-world discrepancies, such as those between RANS and LES models in CFD, can be more complex. To capture these more challenging scenarios, we introduce two new test cases specifically designed to emulate nontrivial LF-HF mismatch beyond simple grid-resolution differences, enabling a more realistic assessment of multi-fidelity strategies under substantial model discrepancies. We find that methods directly feeding LF predictions into HF models are prone to error amplification under large LF-HF discrepancies, particularly in autoregressive settings. In contrast, transfer learning, which uses LF-trained weights to initialise the HF model, consistently improves accuracy by providing a robust prior refined with HF data. These results highlight the importance of establishing transfer learning as a reliable approach for efficient multi-fidelity PDE surrogate modelling.

Problem

• multi-fidelity learning
• neural operators
• PDE surrogate modeling
• fidelity discrepancy
• high-fidelity data scarcity

Innovation

• multi-fidelity learning
• neural operators
• transfer learning
• PDE surrogate modeling
• fidelity discrepancy