# Deep Multi-Fidelity Active Learning of High-dimensional Outputs

arXiv: https://arxiv.org/abs/2012.00901 · HTML: https://mlsanity.com/paper/2012.00901/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2020.

## Built on by (most cited first, 22 of 22)

- [Multi-Resolution Active Learning of Fourier Neural Operators](https://mlsanity.com/paper/2309.16971/) (2023)
- [Batch Multi-Fidelity Active Learning with Budget Constraints](https://mlsanity.com/paper/2210.12704/) (2022)
- [Multi-Fidelity Active Learning with GFlowNets](https://mlsanity.com/paper/2306.11715/) (2023)
- [Multi-Fidelity Residual Neural Processes for Scalable Surrogate Modeling](https://mlsanity.com/paper/2402.18846/) (2024)
- [Disentangled Multi-Fidelity Deep Bayesian Active Learning](https://mlsanity.com/paper/2305.04392/) (2023)
- [A Latent Variable Approach for Non-Hierarchical Multi-Fidelity Adaptive Sampling](https://mlsanity.com/paper/2310.03298/) (2023)
- [Deep Bayesian Active Learning for Accelerating Stochastic Simulation](https://mlsanity.com/paper/2106.02770/) (2021)
- [Infinite-Fidelity Coregionalization for Physical Simulation](https://mlsanity.com/paper/2207.00678/) (2022)
- [Interactive Multi-fidelity Learning for Cost-effective Adaptation of Language Model with Sparse Human Supervision](https://mlsanity.com/paper/2310.20153/) (2023)
- [Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation](https://mlsanity.com/paper/2410.13794/) (2024)
- [Active Learning with Selective Time-Step Acquisition for PDEs](https://mlsanity.com/paper/2511.18107/) (2025)
- [Cost-effective Framework for Gradual Domain Adaptation with Multifidelity](https://mlsanity.com/paper/2202.04359/) (2022)
- [GAR: Generalized Autoregression for Multi-Fidelity Fusion](https://mlsanity.com/paper/2301.05729/) (2023)
- [Diffusion-Generative Multi-Fidelity Learning for Physical Simulation](https://mlsanity.com/paper/2311.05606/) (2023)
- [Aligned Multi Objective Optimization](https://mlsanity.com/paper/2502.14096/) (2025)
- [MFBind: a Multi-Fidelity Approach for Evaluating Drug Compounds in Practical Generative Modeling](https://mlsanity.com/paper/2402.10387/) (2024)
- [MF-LAL: Drug Compound Generation Using Multi-Fidelity Latent Space Active Learning](https://mlsanity.com/paper/2410.11226/) (2024)
- [COMPOL: A Unified Neural Operator Framework for Scalable Multi-Physics Simulations](https://mlsanity.com/paper/2501.17296/) (2025)
- [Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions](https://mlsanity.com/paper/2605.16118/) (2026)
- [Active Learning with Weak Supervision for Gaussian Processes](https://mlsanity.com/paper/2204.08335/) (2022)
- [Differentiable Multi-Fidelity Fusion: Efficient Learning of Physics Simulations with Neural Architecture Search and Transfer Learning](https://mlsanity.com/paper/2306.06904/) (2023)
- [Ensemble models outperform single model uncertainties and predictions for operator-learning of hypersonic flows](https://mlsanity.com/paper/2311.00060/) (2023)

## Abstract (arXiv)

Many applications, such as in physical simulation and engineering design, demand we estimate functions with high-dimensional outputs. The training examples can be collected with different fidelities to allow a cost/accuracy trade-off. In this paper, we consider the active learning task that identifies both the fidelity and input to query new training examples so as to achieve the best benefit-cost ratio. To this end, we propose DMFAL, a Deep Multi-Fidelity Active Learning approach. We first develop a deep neural network-based multi-fidelity model for learning with high-dimensional outputs, which can flexibly, efficiently capture all kinds of complex relationships across the outputs and fidelities to improve prediction. We then propose a mutual information-based acquisition function that extends the predictive entropy principle. To overcome the computational challenges caused by large output dimensions, we use multi-variate Delta's method and moment-matching to estimate the output posterior, and Weinstein-Aronszajn identity to calculate and optimize the acquisition function. The computation is tractable, reliable and efficient. We show the advantage of our method in several applications of computational physics and engineering design.