Quantum computers have been proposed as a solution for efficiently solving non-linear differential equations (DEs), a fundamental task across diverse technological and scientific domains. However, a crucial milestone in this regard is to design protocols that are hardware-aware, making efficient use of limited available quantum resources. We focus here on promising variational methods derived from scientific machine learning: differentiable quantum circuits (DQC), addressing specifically their cost in number of circuit evaluations. Reducing the number of quantum circuit evaluations is particularly valuable in hybrid quantum/classical protocols, where the time required to interface and run quantum hardware at each cycle can impact the total wall-time much more than relatively inexpensive classical post-processing overhead. Here, we propose and test two sample-efficient protocols for solving non-linear DEs, achieving exponential savings in quantum circuit evaluations. These protocols are based on redesigning the extraction of information from DQC in a ``measure-first" approach, by introducing engineered cost operators similar to the randomized-measurement toolbox (i.e. classical shadows). In benchmark simulations on one and two-dimensional DEs, we report up to $\sim$ 100 fold reductions in circuit evaluations. Our protocols thus hold the promise to unlock larger and more challenging non-linear differential equation demonstrations with existing quantum hardware.
NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations
Department of Mathematics, University of California Riverside（加州大学河滨分校数学系）；Department of Mechanical Engineering, University of California Riverside（加州大学河滨分校机械工程系）；German Innovation Center, Procter & Gamble Service GmbH（普罗cter与 gamble 服务德国创新中心）
Large-scale Neural Solvers for Partial Differential Equations
Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗森多夫亥姆霍兹中心）；Technische Universität Dresden（德累斯顿工业大学）；Center for Advanced Systems Understanding (CASUS)（高级系统理解中心）
Opening the Blackbox: Accelerating Neural Differential Equations by Regularizing Internal Solver Heuristics
Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）；Julia Computing；Massachusetts Institute of Technology（麻省理工学院）；University of Maryland Baltimore（马里兰大学巴尔的摩分校）；Pumas AI
Engineering application of physics-informed neural networks for Saint-Venant torsion
Department of Quantum System Engineering, Jeonbuk National University（全量子系统工程系，全州国立大学）；Department of Mathematics, Inha University（数学系，inha大学）；Electrified Propulsion Structure CAE Team, Hyundai Motor Company（电动推进结构CAE团队，现代汽车公司）；Department of Mechanical Engineering, Inha University（机械工程系，inha大学）；Graduate School of Integrated Energy-AI, Jeonbuk National University（整合能源-人工智能研究生院，全州国立大学）
A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method
Technical University of Darmstadt（达姆施塔特工业大学）；RWTH Aachen University（亚琛工业大学）
Deep Parallel Spectral Neural Operators for Solving Partial Differential Equations with Enhanced Low-Frequency Learning Capability
College of Computer Science and Technology（计算机科学与技术学院）；China University of Petroleum (East China)（中国石油大学（华东））；State Key Laboratory of Chemical Safety（化学安全国家重点实验室）
Efficient lattice field theory simulation using adaptive normalizing flow on a resistive memory-based neural differential equation solver
the University of Hong Kong（香港大学）；Southern University of Science and Technology（南方科技大学）；ACCESS - AI Chip Center for Emerging Smart Systems, InnoHK Centers, Hong Kong Science Park（ACCESS-新兴智能系统AI芯片中心，InnoHK中心，香港科学园）；Huazhong University of Science and Technology（华中科技大学）
