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Abstract: Multi-fidelity learning involves using training examples at different fidelities or resolutions. High-fidelity examples are of high-quality but often are much more costly to collect than inaccurate, low-fidelity examples. How to retrieve and leverage examples at multiple fidelities is the key to reduce the learning cost while maximizing the efficiency. This talk will introduce our recent work in multi-fidelity learning and active learning for scientific machine learning. Physical simulation is a central task in science and engineering domains. However, traditional numerical methods are known to be computationally costly and lack generalizability across different problems. Learning data-driven surrogate models presents a promising strategy for cost reduction. I will discuss three multi-fidelity, multi-resolution active learning approaches designed to dynamically acquire simulation examples and their fidelities, enhancing surrogate learning performance while minimizing data acquisition costs. I will demonstrate the efficacy of these methods through applications in standard benchmarks of physical simulation and topology structure optimization. Furthermore, I will introduce an infinite fidelity surrogate learning framework, which extends the traditional finite fidelity space to an infinite continuous space. This framework offers exciting possibilities for advancing scientific machine learning by enabling more comprehensive representations of complex systems and phenomena.