Multi-fidelity learning model predicts fine-scale solutions for multiple PDEs
In-Context Learning for Multi-Fidelity Multiscale Model Closure and Cross-Operator Out-Of-Distribution Generalization
Machine Learning
Summary
Solving complicated equations that describe physical systems often requires both simple coarse models and detailed fine models, but combining these can be expensive and slow. The authors created a single learning model that uses a few examples from different equations to quickly predict detailed solutions from coarse inputs for many types of problems. This model can adapt to new equations with minimal extra data and no retraining, saving time and resources. The approach works even when predicting beyond previously seen timeframes, making it flexible for practical use.
What this means in practice
- •For engineering simulation teams: Generate high-accuracy detailed model predictions for different PDE systems using few examples without retraining.
- •For computer graphics developers: Produce detailed multiscale physical effects from coarse simulations efficiently in graphics software pipelines.
Authors
Xinyuan Li, Matthew Zahr, Panos Stinis, Zecheng Zhang
Abstract
Multi-fidelity learning uses low-fidelity information (coarse-scale PDE models) together with limited high-fidelity data (fine-scale solutions) to obtain high-accuracy PDE solutions. It is a data-driven approach to downscale reduced-order models to solve PDEs efficiently and accurately. Standard multi-fidelity learning focuses on one operator or PDE; given a new PDE, retraining using additional coarse-fine solution pairs may be required, which is costly. In this work, we propose an in-context multi-fidelity learning framework for learning multiple multi-fidelity operators with one shared model. The model is trained using paired low- and high-fidelity examples as demonstrations from multiple operators, together with a new coarse-scale solution query, to predict the corresponding fine-scale solution. Once trained, given a new PDE operator, a few (fewer than five) coarse-fine demonstration pairs, and a low-cost coarse query, the model predicts the corresponding fine-scale solution without fine-tuning. When the target operator appears in the training dataset but has a different prediction goal, such as predicting beyond the training time interval, no additional demonstrations are needed. Instead, the model uses training coarse-fine pairs from that PDE and information from other training operators. Testing then only requires generating a low-cost coarse-scale solution query, without fine-tuning. Notably, our smallest model contains only 255,364 trainable parameters, corresponding to approximately 456 parameters per training operator for 560 operators.