# Bridging quantum and classical computing for partial differential equations through multifidelity machine learning

arXiv: https://arxiv.org/abs/2512.05241 · HTML: https://mlsanity.com/paper/2512.05241/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 4 Dec 2025.

## Builds on (most cited first, 15 of 15)

- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities](https://mlsanity.com/paper/2102.13403/) (2021)
- [Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONets](https://mlsanity.com/paper/2204.00997/) (2022)
- [On transfer learning of neural networks using bi-fidelity data for uncertainty propagation](https://mlsanity.com/paper/2002.04495/) (2020)
- [Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2106.13361/) (2021)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [A multifidelity approach to continual learning for physical systems](https://mlsanity.com/paper/2304.03894/) (2023)
- [A Multifidelity deep operator network approach to closure for multiscale systems](https://mlsanity.com/paper/2303.08893/) (2023)
- [Multifidelity Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.14764/) (2024)
- [Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations](https://mlsanity.com/paper/2303.11577/) (2023)

## Built on by (most cited first, 1 of 1)

- [Conformalized Quantum DeepONet Ensembles: Towards Scalable Operator Learning with Distribution-Free Guarantees](https://mlsanity.com/paper/2605.00330/) (2026)

## Abstract (arXiv)

Quantum algorithms for partial differential equations (PDEs) face severe practical constraints on near-term hardware: limited qubit counts restrict spatial resolution to coarse grids, while circuit depth limitations prevent accurate long-time integration. These hardware bottlenecks confine quantum PDE solvers to low-fidelity regimes despite their theoretical potential for computational speedup. We introduce a multifidelity learning framework that corrects coarse quantum solutions to high-fidelity accuracy using sparse classical training data, facilitating the path toward practical quantum utility for scientific computing. The approach trains a low-fidelity surrogate on abundant quantum solver outputs, then learns correction mappings through a multifidelity neural architecture that balances linear and nonlinear transformations. Demonstrated on benchmark nonlinear PDEs including viscous Burgers equation and incompressible Navier-Stokes flows via quantum lattice Boltzmann methods, the framework successfully corrects coarse quantum predictions and achieves temporal extrapolation well beyond the classical training window. This strategy illustrates how one can reduce expensive high-fidelity simulation requirements while producing predictions that are competitive with classical accuracy. By bridging the gap between hardware-limited quantum simulations and application requirements, this work establishes a pathway for extracting computational value from current quantum devices in real-world scientific applications, advancing both algorithm development and practical deployment of near-term quantum computing for computational physics.