Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes (2024)

I Introduction

Computational fluid dynamics (CFD) solvers are primarily used to find the distribution of the velocity vector, $\bm{v}$, and pressure, $p$, of a fluid (or several fluids) given the initial conditions (e.g., initial velocity profile) and the geometrical domain in which the fluid flowscfd_review ; Anderson1995ComputationalFD .To do this, it is necessary to solve a system of differential equationsSimmons1972DifferentialEW called the Navier-Stokes (NS) equations that govern the fluid flowJameson2009MeshlessMF . A well-established approach is to use numerical CFD solvers from several vendors, as well as publicly accessible alternatives, such as OpenFOAMopenfoam or Ansysansys . These solvers discretize a given fluid volume into several small parts known as cellsUnstructured , where it is easier to get an approximate solution and then join the solutions of all the cells to get a complete distribution of the pressure and velocity over the entire geometrical domain.

While this is a rather crude explanation of how CFD solvers work, discretizing a large domain into smaller pieces accurately captures one of their main working principlesMari2013voFoamA .The runtime of the computation and the accuracy of the solution both sensitively depend on the fineness of discretization, with finer grids taking longer but giving more accurate solutions.Furthermore, any changes to the geometrical parameters necessitate the creation of a new mesh and a new simulation. This process consumes both time and resources since one has to remesh and rerun the simulation every time a geometrical parameter is alteredpinn_ns_review .

We propose a workflow employing physics-informed neural networks (PINNs)raissi2019physics to escape the need to restart the simulations whenever a geometrical property is changed completely. A PINN is a new promising tool for solving all kinds of parameterized partial differential equations (PDEs)pinn_ns_review , as it does not require many prior assumptions, linearization or local time-stepping. One defines an architecture of the neural network (number of neurons, layers, etc.) and then embeds physical laws and boundary conditions into it via constructing an appropriate loss function, so the prior task immediately travels to the domain of optimization problems.

For the classical solver, in the case of a parameterized geometric domain problem, getting accurate predictions for new modified shapes requires a complete program restart, even if the geometry has changed slightly. In the case of a PINN, to overcome this difficulty, one can use the transfer learning methodTransfer_Learning (Sec.III.1), which allows a model previously trained on some geometry to be trained on a slightly modified geometry without the need for a complete reset.

Also, using a trained PINN, it is easy to obtain a solution for other parameters of the PDE equation (e.g. kinematic viscosity in the NS equationJameson2009MeshlessMF , thermal conductivity in the heat equation, etc.) with no additional training or restart of the neural network, but in the case of traditional solvers, restarts cannot be avoided.

One of the features that makes PINNs appealing is that they suffer less from the curse of dimensionality. Finite discretization of a $d$-dimensional cube with $N$ points along each axis would require $N^{d}$ points for a traditional solver. In other words, the complexity of the problem grows exponentially as the sampling size $d$ increases. Using a neural network, however, one can define a $\mathbb{R}^{d}\rightarrow\mathbb{R}$ mapping (in case of just one target feature) with some weight parameters. Research on the topic suggests that the amount of weights/complexity of a problem in such neural networks grows polynomially with the input dimension $d$hutzenthaler2020proof ; grohs2018proof . This theoretical foundation alone allows PINNs to be a competitive alternative to solvers.

It is worth noting that although a PINN does not require $N^{d}$ points for inference, it does require many points for training. There exist a variety of sampling methods, such as latin hypercube samplingstein1987large , Sobol sequencessobol2011construction , etc., which can be used to improve a PINN’s convergence on trainingzubov2021neuralpde . However, a simple static grid of points is used in this work for the purpose of simplicity.

Recent developments in automatic differentiation enable us to compute the exact derivatives of any order of a PINN, so there is no need to use finite differences or any other approximate differentiation techniques. It, therefore, seems that we do not require a discretized mesh over the computational domain.However, we still need a collection of points from the problem’s domain to train and evaluate a PINN.For a PINN to provide an accurate solution for a fluid dynamics problem, it is important to have a high expressivity (ability to learn solutions for a large variety of, possibly complex, problems).Fortunately, expressivity is a known strength of quantum computersmo_2022 ; schuld2021effect ; schuld2020circuit . Furthermore, quantum circuits are differentiable, meaning their derivatives can be calculated analytically, which is essential for noisy intermediate-scale quantum devices.

In this article, we propose a hybrid quantum PINN (HQPINN) shown in Fig.1 to solve the NS equations with a steady flow in a 3D $Y$-shape mixer. The general principles and loss function of PINN workflow are described in Sec.II. The problem description, including the geometrical details, is presented in Sec.III while in Sec.III.0.5 and Sec.III.2, we describe classical and hybrid PINNs in detail. Sec.III.0.1 explains the intricacies of PINN’s training process and simulation results. A transfer learning approach, applied to PINNs, is presented in Sec.III.1.Conclusions and further plans are described in Sec.IV.

II Physics-informed neural networks for solving partial differential equations

Let us consider a neural network $u(\bm{r},t)$ that takes coordinates, $\bm{r}$, and time, $t$, as input and yields some real value (e.g., the pressure of a liquid at this coordinate at this particular moment).

We can evaluate $u(\bm{r},t)$ at any point in the computational domain via a forward pass and compute its derivatives (of any order) $\partial_{t}^{n}u(\bm{r},t)$, $\partial_{\bm{r}}^{n}u(\bm{r},t)$ through backpropagationrumelhart1986learning . Therefore, we could substitute $f(\bm{r},t)=u(\bm{r},t)$ and try to learn the correct solution for the PDE via common machine learning gradient optimization methodsgradient (e.g., gradient descent).

This approach is inspired firstly by the ability to calculate the exact derivatives of a neural network via autodifferentiationautodiffPaper and secondly by neural networks being universal function approximatorsHornik1989MultilayerFN .

III Simulations

In this work, we consider the steady (i.e., time-independent) flow of an incompressible fluid in 3D without any external forces.

III.0.2 Cylinder flow simulation

At first, we used a simple 3D cylinder flow as a baseline solution for classical PINN. We also validated PINN’s predictions by comparing them to OpenFOAM’s solution.

In this simulation, we have no-slip boundary condition on the walls, fixed velocity of $v_{0}=10\ \text{mm}/\text{s}$ on the inlet, and fixed zero pressure $p=0\ \text{Pa}$ on the outlet.The fluid is water with standard density $\rho=1\ \text{g}/\text{cm}^{3}$ and $\nu=1\ \text{mm}^{2}/\text{s}$ kinematic viscosity.The cylinder has radius of $2\ \text{mm}$ and height of $10\ \text{mm}$ (2). Reynolds number for such flow is $10$, so it can be considered laminar.

The training was done on a single NVIDIA A100 GPU for $20^{\prime}000$ iterations with an L-BFGS optimizer. For its training, the model uses $25^{\prime}000$ points inside the cylinder, which are taken straight from the mesh nodes used in OpenFOAM’s simulation.PINN represented the solver’s solution quite well: the relative error of velocity magnitude averaged over the whole cylinder is $1.2\ \%$, and the mean relative error of pressure is $0.8\ \%$. Refer to Fig.3 for ground truth (OpenFOAM) and prediction (PINN) field values, as well as the distribution of relative error across the geometry.

The only error spikes both for pressure and velocity fields are located near the inlet edge of the pipe, where the uniform velocity profile of $10\ \text{mm}/\text{s}$ abruptly changes to $0$ due to non-slip boundary condition on walls, which is somewhat unphysical. This can be corrected by making the inlet velocity profile parabolic instead of uniform, but we decided to stick to a simple problem statement for the baseline solution.

III.0.3 Generalization of physics-informed neural networks

To check if our baseline model possesses any generalization capabilities, we trained the same model as in SectionIII.0.2, but now incorporating $4$ input parameters: the $x,y,z$ coordinates and the kinematic viscosity $\nu$, which was held constant at $1\,\text{mm}^{2}/\text{s}$ for the aforementioned model. Given that we employ a full-batch strategy, where the optimizer processes the gradient of all points in the geometry simultaneously, such a modification significantly increases the memory requirements for storing the total gradient. Consequently, we limited our training set to a set of $\nu$ values, $\{1,2,3,4\}\,\text{mm}^{2}/\text{s}$. Nonetheless, we believe this range is sufficient to explore the generalization capabilities of the classical PINN.

Following the same training regime as in SectionIII.0.2, we present the results on the test data in Table1. The results indicate that the model generalizes well across the range of $\nu$ values it trained on.

$\nu,\ \text{mm}^{2}/\text{s}$	$p$ rel. error, %	$\|\bm{v}\|$ rel. error, %
$1$	$10.0$	$4.7$
$\bm{1.5}$	$2.0$	$2.8$
$2$	$2.7$	$1.5$
$\bm{2.5}$	$5.9$	$3.6$
$3$	$8.5$	$5.8$
$\bm{3.5}$	$10.5$	$7.7$
$4$	$12.3$	$9.4$
$\bm{5}$	$15.1$	$12.3$
$\bm{10}$	$21.0$	$24.6$

However, when the model attempts to extrapolate solutions for $\nu$ values outside its training range, the quality of the solutions begins to degrade rapidly. More specifically, the solution tends to converge to the trivial case of $\bm{v}=0$ as it approaches the end of the pipe. This phenomenon will be revisited and discussed further in SectionIII.0.4, where we attempt simulations on more complex geometries than the current one.

III.0.4 Y-shaped mixer flow simulation

This time, we try to simulate the flow of liquid in a $Y$-shaped mixer consisting of three tubes (Fig.4). The mixing fluids are identical and have parameters $\rho=1.0\ \text{kg}/\text{m}^{3}$ and $\nu=1.0\ \text{m}^{2}/\text{s}$.

The imposed boundary conditions are as follows:

• •

  no-slip BC on walls: $\bm{v}(\bm{r})|_{\text{walls}}=0$,

• •

  fixed velocity profile on inlets: $\bm{v}(\bm{r})|_{\text{inlets}}=\bm{v}_{0}(\bm{r})$,

• •

  fixed pressure on outlets: $p(\bm{r})|_{\text{outlets}}=p_{0}$,

where, $\bm{v}_{0}(\bm{r})$ is a parabolic velocity profile on each inlet and $p_{0}(\bm{r})=0$.

The PINN was trained via full-batch gradient descent with the Adam optimizeradam for $1000$ epochs and then with the L-BFGS optimizer for $100$ epochs until gradient vanished and training became impossible. After the training, the PINN managed to learn a non-trivial downward flow at the beginning of both pipes. On the edges of these pipes, the velocities become zero, as they should, due to no-slip wall boundary conditions. However, further down the mixer, the solution degenerates to zero, so it does not even reach the mixing point. This fact should at least violate the continuity equation because there is an inward flow without matching outward flow. This problem is primarily caused by gradient vanishing inherent to PINN models.

In an attempt to overcome the challenge, we introduce an HQPINN model in III.2, which shows better results in terms of PDE and boundary conditions satisfaction. We also research potential ways of making a generalizable PINN in III.1 by employing a transfer learning method.

III.2 Hybrid quantum PINN

Quantum machine learning was shown to be applicable to solving differential equationsquantum_expressivity ; paine2023physicsinformed ; PhysRevA.107.032428 . Here, we introduce a hybrid quantum neural network called HQPINN and compare it against its classical counterpart, classical PINN. As shown in Fig.1, the architecture of HQPINN is comprised of classical fully connected layers, specifically a multilayer perceptron, coupled with a Parallel Hybrid Networkkordzanganeh2023parallel . The latter is a unique intertwining of a Quantum depth-infused layersag_hyb_2022 ; schuld2021effect and a classical layer. Interestingly, the first 15 units ($\phi_{1},\dots,\phi_{15}$, depicted in green) of the layer are dedicated to the quantum layer, whereas the information from one orange unit proceeds along the classical pathway. This parallel structure enables simultaneous information processing, thereby enhancing the efficiency of the learning process.

Regarding the quantum depth-infused layer, it is implemented as a variational quantum circuit (VQC), a promising strategy for navigating the complexities of the Noisy Intermediate-Scale Quantum (NISQ) epochpreskill2018quantum . The NISQ era is characterized by quantum devices with a limited number of qubits that cannot yet achieve error correction, making strategies like VQCs particularly relevantzhao2019qdnn ; dou2021unsupervised ; sag_hyb_2022 .

The capacity of HQPINN to direct diverse segments of the input vector toward either the quantum or classical part of the network equips PINNs with an enhanced ability to process and learn from various patterns more efficiently. For instance, some patterns may be optimally processed by the quantum network, while others might be better suited for the classical network. This flexibility in processing contributes to the robust learning capabilities of HQPINN.

III.2.1 Quantum Depth-Infused Layer

Transitioning into the building blocks of our model, the quantum depth-infused layer takes center stage. Quantum gates are the basic building blocks for any quantum circuit, including those used for machine learning. Quantum gates come in single-qubit (e.g., rotation gate $R_{y}(\theta)$, gate that plays a central role in quantum machine learning) and multiple-qubit gates (e.g., CNOT) and modulate the state of qubits to perform computations. The $R_{y}(\theta)$ gate rotates the qubit around the y-axis of the Bloch sphere by an angle $\theta$, while the two-qubit CNOT gate changes the state of one qubit based on the current state of another qubit. Gates can be fixed, which means they perform fixed calculations, such as the Hadamard gate, or they can be variable, such as the rotation gate that depends on the rotation angle and may perform computations with tunable parameters.

To extract the results, qubits are measured and projected onto a specific basis, and the expected value is calculated. When using the $\sigma_{z}$ Pauli matrix observable, the expected value of the measurement is represented as: $\left\langle\psi\right|\sigma_{z}\left|\psi\right\rangle$, with $\psi$ signifying the wave function that depicts the current state of our quantum system. For a more detailed understanding of quantum circuits, including logic gates and measurements, standard quantum computing textbooks such asnielsen2002quantum offer a comprehensive guide.

To refine the functioning of the network we propose, the initial phase of data processing is performed on a classical computer. Subsequently, these data are integrated into the quantum gate parameters in an encoding layer (blue gates) repeated $d=5$ times, forming part of the quantum depth-infused layer followed by the variational layer (green gates). Number of repetitions of the variational layer $m=2$. As the algorithm develops, these gate parameters in the variational layer are dynamically adjusted. Finally, at the measurement level, the qubits are quantified, leading to a series of classical bits as the output.

Conceptually, a quantum algorithm resembles a black box that receives classical information as input and emits classical information as output. The objective here is to fine-tune the variational parameters such that the measurement outcome most accurately reflects the prediction function. In essence, this parameter optimization is akin to optimizing the weights in a classical neural network, thereby effectively training the quantum depth-infused layer.

III.2.2 Training hybrid quantum physics-informed neural network

The HQPINN consists of the classical PINN with weights pre-initialized from the previous stage (as described in Sec.III.0.5), a parallel hybrid network and a fully-connected layer at the end.

The training process of the hybrid PINN does not differ from that of the classical PINN except in the following ways. Firstly, all calculations are done on the classical simulator of quantum hardware, the QMware serverqmw_qmw_2022 , which has recently been shown to be quite good for running hybrid algorithmsmo_bench_2022 .

Secondly, how does one backpropagate through the quantum circuit layer? The answer is to use the “adjoint differentiation” method, introduced inadjointdiff , which helps to compute derivatives of a VQC on a classical simulator efficiently.

This time the model was trained for $100$ epochs using mini-batch gradient descent with the Adam optimizer (Fig.5). The mini-batch strategy was employed due to the very low training speed of quantum circuits, as they train on a CPU. We will then compare this model with a purely classical one, with the same architecture from Sec.III.0.5, but this time trained only with mini-batch Adam. All learning hyperparameters (learning rate, scheduler parameters, batch size) are shared between the quantum and classical models. Comparing the two, Fig.6 shows that the quantum model outperforms the classical one in terms of the loss value by $21\%$. As the loss function for PINNs directly corresponds to PDE and BC satisfaction, it implies that HQPINN has achieved better physical accuracy before the gradient vanish. However, the mini-batch training strategy for the HQPINN is far from ideal and stems from the large training time of simulated quantum circuits. Proper hardware backend for highly parallel GPU-accelerated quantum computing could greatly extend the HQPINN advantage.

Additionally, the question of computational demands scaling is of importance. For the CFD examples considered in the paper, we used a three-qubit circuit, which is not computationally demanding to execute on a classical computer. However, with each additional qubit, the runtime of the quantum circuit execution will approximately double. That means, according to the benchmarksmo_bench_2022 , the runtime will become significant beyond 20 qubits circuits. Nonetheless, with the development of quantum computers, the scaling is expected to be more favorable as the runtime on a quantum chip would stay constant in case the circuit depth is fixed. Therefore, we believe that for more complex shapes, the simulation would require more data and, hence, larger HQPINN models require the need of quantum computers for simulation.

IV Discussion

In our investigation, we pursued two distinct goals aimed at advancing the capabilities of PINNs through the integration of quantum computing methodologies. The first objective focused on enhancing the performance of classical PINNs in solving PDEs within 3D geometries through the adoption of HQPINNs. This endeavor was motivated by the recognized success of classical PINNs in 2D problem settingsraissi2019physics and the existing gap in their application to more complex 3D scenarios. The significance of this goal lies in its potential to broaden the applicability of PINNs to a wider range of scientific and engineering problems characterized by three-dimensional spaces.

Our second goal was to investigate the presence and feasibility of transfer learning in classical PINNs when applied to 3D geometries. This exploration is crucial for understanding the ability of PINNs to generalize across different geometrical configurations and physical problems, potentially enabling more efficient re-training processes on novel problems and geometries. The overarching aim here is to enhance the model’s versatility and reduce computational costs associated with the training of neural networks for new tasks.

Our results in applying HQPINNs to 3D PDE problems demonstrate a notable improvement in reducing PDE loss, marking a significant step forward in our first objective. Specifically, we observed a $21\%$ reduction in loss when comparing the performance of HQPINNs against their purely classical counterparts. This quantitative improvement highlights the enhanced expressiveness and computational efficiency brought about by the integration of quantum layers, suggesting that quantum computing elements can indeed augment the capability of PINNs in handling the intricacies of 3D problems. However, it is important to recognize that despite these advancements, achieving an optimal solution for 3D PDEs remains a challenging endeavor. The observed loss reduction, while substantial, does not resolve the complexities of 3D geometrical problem solving, indicating the need for further refinement and exploration of the HQPINN framework.

In the realm of transfer learning, our exploration yielded encouraging signs that classical PINNs possess an inherent capability to adapt to variations in a geometrical shape, when theangle of a $Y$-shaped mixer was changed. This qualitative finding is instrumental in demonstrating the potential for PINNs to be applied in shape optimization tasks and other applications requiring flexibility across different geometrical configurations. However, the journey towards realizing full transfer learning and generalization capabilities in PINNs is still in its early stages. Our initial successes serve as a foundation for further research, underscoring the necessity for continued development and investigation into the mechanisms that enable effective transfer learning within PINNs.

In developing our HQPINNs, we have also considered recent advances in other quantum and quantum-inspired methods for solving nonlinear equations, such as based on the matrix product states approachgourianov2022quantum and variational quantum algorithms for nonlinear problemslubasch2020variational ; quantum_expressivity . While our approach primarily focuses on handling complex boundary conditions, the integration of these quantum-inspired methods and variational quantum algorithms could further improve the performance and accuracy of our proposed quantum PINN model.

Collectively, our findings contribute valuable insights into the potential of hybrid quantum neural networks and the prospects for transfer learning in solving complex physical problems. As we continue to push the boundaries of what is achievable with PINNs, our future efforts will focus on refining these models to approach accurate full-scale CFD simulation in complex 3D shapes.

The plan includes exploring better architectures for the quantum PINN, investigating their impact on expressiveness, generalizability and optimization landscape, and trying data-driven approaches. Entirely different networks, such as neural operatorsneural_ops ; fno and graph neural networksmesh_based_gnn ; sanchez2020learning , could also be considered in a quantum setting and enhanced with quantum circuits.

References

• (1)MohdHafiz Zawawi, ASaleha, ASalwa, NHHassan, NazirulMubin Zahari,MohdZakwan Ramli, and ZakariaChe Muda.A review: Fundamentals of computational fluid dynamics (CFD).In AIP conference proceedings. AIP Publishing LLC, 2018.
• (2)JohnDavid Anderson and John Wendt.Computational fluid dynamics, volume 206.Springer, 1995.
• (3)GeorgeFinlay Simmons.Differential equations with applications and historical notes.In Differential Equations With Applications and HistoricalNotes. McGraw-Hill, 1972.
• (4)AaronJon Katz.Meshless methods for computational fluid dynamics.Stanford University, 2009.
• (5)OpenFOAM.https://www.openfoam.com/, 2022.
• (6)Ansys.https://www.ansys.com/, 2022.
• (7)Tomislav Marić, DouglasB. Kotheb, and Dieter Bothe.Unstructured un-split geometrical volume-of-fluid methods - Areview.Journal of Computational Physics, 420, 2020.
• (8)Tomislav Marić, Holger Marschall, and Dieter Bothe.vofoam - a geometrical volume of fluid algorithm on arbitraryunstructured meshes with local dynamic adaptive mesh refinement usingOpenFOAM.arXiv preprint arXiv: 1305.3417, 2013.
• (9)Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and GeorgeEmKarniadakis.Physics-informed neural networks (PINNs) for fluid mechanics: Areview.Acta Mechanica Sinica, pages 1–12, 2022.
• (10)Maziar Raissi, Paris Perdikaris, and GeorgeE Karniadakis.Physics-informed neural networks: A deep learning framework forsolving forward and inverse problems involving nonlinear partial differentialequations.Journal of Computational Physics, 378:686–707, 2019.
• (11)Behnam Neyshabur, Hanie Sedghi, and Chiyuan Zhang.What is being transferred in transfer learning?arXiv preprint arXiv:2008.11687, 2020.
• (12)Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, and TuanAnh Nguyen.A proof that rectified deep neural networks overcome the curse ofdimensionality in the numerical approximation of semilinear heat equations.SN partial differential equations and applications, 1:1–34,2020.
• (13)Philipp Grohs, Fabian Hornung, Arnulf Jentzen, and Philippe VonWurstemberger.A proof that artificial neural networks overcome the curse ofdimensionality in the numerical approximation of Black-Scholes partialdifferential equations.arXiv preprint arXiv:1809.02362, 2018.
• (14)Michael Stein.Large sample properties of simulations using latin hypercubesampling.Technometrics, 29(2):143–151, 1987.
• (15)IlyaM Sobol’, Danil Asotsky, Alexander Kreinin, and Sergei Kucherenko.Construction and comparison of high-dimensional Sobol’generators.Wilmott, 2011(56):64–79, 2011.
• (16)Kirill Zubov, Zoe McCarthy, Yingbo Ma, Francesco Calisto, Valerio Pagliarino,Simone Azeglio, Luca Bottero, Emmanuel Luján, Valentin Sulzer, AshutoshBharambe, etal.Neuralpde: Automating physics-informed neural networks (PINNs) witherror approximations.arXiv preprint arXiv:2107.09443, 2021.
• (17)Frank Gaitan.Finding flows of a Navier–Stokes fluid through quantumcomputing.npj Quantum Information, 6(1):1–6, 2020.
• (18)Vedran Dunjko and HansJ Briegel.Machine learning & artificial intelligence in the quantum domain: areview of recent progress.Reports on Progress in Physics, 81(7):074001, 2018.
• (19)Alexey Melnikov, Mohammad Kordzanganeh, Alexander Alodjants, and Ray-Kuang Lee.Quantum machine learning: from physics to software engineering.Advances in Physics: X, 8(1):2165452, 2023.
• (20)Hartmut Neven, VasilS. Denchev, Geordie Rose, and WilliamG. Macready.QBoost: Large scale classifier training with adiabatic quantumoptimization.In Steven C.H. Hoi and Wray Buntine, editors, Proc. Asian Conf.Mach. Learn., volume25 of Proceedings of Machine Learning Research,pages 333–348. PMLR, 2012.
• (21)Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd.Quantum support vector machine for big data classification.Physical Review Letters, 113:130503, Sep 2014.
• (22)Valeria Saggio, BeateE Asenbeck, Arne Hamann, Teodor Strömberg, PeterSchiansky, Vedran Dunjko, Nicolai Friis, NicholasC Harris, Michael Hochberg,Dirk Englund, etal.Experimental quantum speed-up in reinforcement learning agents.Nature, 591(7849):229–233, 2021.
• (23)MoKordzanganeh, Daria Kosichkina, and Alexey Melnikov.Parallel hybrid networks: an interplay between quantum and classicalneural networks.Intelligent Computing, 2:0028, 2023.
• (24)Andrii Kurkin, Jonas Hegemann, MoKordzanganeh, and Alexey Melnikov.Forecasting the steam mass flow in a powerplant using the parallelhybrid network.arXiv preprint arXiv:2307.09483, 2023.
• (25)Michael Perelshtein, Asel Sagingalieva, Karan Pinto, Vishal Shete, AlexeyPakhomchik, Artem Melnikov, Florian Neukart, Georg Gesek, Alexey Melnikov,and Valerii Vinokur.Practical application-specific advantage through hybrid quantumcomputing.arXiv preprint arXiv:2205.04858, 2022.
• (26)Asel Sagingalieva, Mohammad Kordzanganeh, Nurbolat Kenbayev, Daria Kosichkina,Tatiana Tomashuk, and Alexey Melnikov.Hybrid quantum neural network for drug response prediction.Cancers, 15(10):2705, 2023.
• (27)A.I. Gircha, A.S. Boev, K.Avchaciov, P.O. Fedichev, and A.K. Fedorov.Training a discrete variational autoencoder for generative chemistryand drug design on a quantum annealer.arXiv:2108.11644, 2021.
• (28)Serge Rainjonneau, Igor Tokarev, Sergei Iudin, Saaketh Rayaprolu, Karan Pinto,Daria Lemtiuzhnikova, Miras Koblan, Egor Barashov, MoKordzanganeh, MarkusPflitsch, and Alexey Melnikov.Quantum algorithms applied to satellite mission planning for Earthobservation.IEEE Journal of Selected Topics in Applied Earth Observationsand Remote Sensing, 16:7062–7075, 2023.
• (29)Asel Sagingalieva, Andrii Kurkin, Artem Melnikov, Daniil Kuhmistrov, etal.Hybrid quantum ResNet for car classification and its hyperparameteroptimization.Quantum Machine Intelligence, 5(2):38, 2023.
• (30)Nathan Haboury, MoKordzanganeh, Sebastian Schmitt, Ayush Joshi, Igor Tokarev,Lukas Abdallah, Andrii Kurkin, Basil Kyriacou, and Alexey Melnikov.A supervised hybrid quantum machine learning solution to theemergency escape routing problem.arXiv preprint arXiv:2307.15682, 2023.
• (31)Javier Alcazar, Vicente Leyton-Ortega, and Alejandro Perdomo-Ortiz.Classical versus quantum models in machine learning: insights from afinance application.Machine Learning: Science and Technology, 1(3):035003, 2020.
• (32)Brian Coyle, Maxwell Henderson, Justin ChanJin Le, Niraj Kumar, Marco Paini,and Elham Kashefi.Quantum versus classical generative modelling in finance.Quantum Science and Technology, 6(2):024013, 2021.
• (33)Marco Pistoia, SyedFarhan Ahmad, Akshay Ajagekar, Alexander Buts, ShouvanikChakrabarti, Dylan Herman, Shaohan Hu, Andrew Jena, Pierre Minssen, PradeepNiroula, Arthur Rattew, Yue Sun, and Romina Yalovetzky.Quantum Machine Learning for Finance.arXiv preprint arXiv:2109.04298, 2021.
• (34)Dimitrios Emmanoulopoulos and Sofija Dimoska.Quantum Machine Learning in Finance: Time Series Forecasting.arXiv preprint arXiv:2202.00599, 2022.
• (35)ElAmine Cherrat, Snehal Raj, Iordanis Kerenidis, Abhishek Shekhar, Ben Wood,Jon Dee, Shouvanik Chakrabarti, Richard Chen, Dylan Herman, Shaohan Hu,Pierre Minssen, Ruslan Shaydulin, Yue Sun, Romina Yalovetzky, and MarcoPistoia.Quantum Deep Hedging.arXiv preprint arXiv:2303.16585, 2023.
• (36)Arsenii Senokosov, Alexandr Sedykh, Asel Sagingalieva, Basil Kyriacou, andAlexey Melnikov.Quantum machine learning for image classification.Machine Learning: Science and Technology, 5(1):015040, 2024.
• (37)Wei Li, Peng-Cheng Chu, Guang-Zhe Liu, Yan-Bing Tian, Tian-Hui Qiu, and Shu-MeiWang.An Image Classification Algorithm Based on Hybrid Quantum ClassicalConvolutional Neural Network.Quantum Engineering, 2022:1–9, 2022.
• (38)ANaumov, ArMelnikov, VAbronin, FOxanichenko, KIzmailov, MPflitsch,AMelnikov, and MPerelshtein.Tetra-AML: Automatic machine learning via tensor networks.arXiv preprint arXiv:2303.16214, 2023.
• (39)Farina Riaz, Shahab Abdulla, Hajime Suzuki, Srinjoy Ganguly, RavineshC. Deo,and Susan Hopkins.Accurate Image Multi-Class Classification Neural Network Model withQuantum Entanglement Approach.Sensors, 23(5):2753, 2023.
• (40)Zhenhou Hong, Jianzong Wang, Xiaoyang Qu, Chendong Zhao, Wei Tao, and JingXiao.QSpeech: Low-Qubit Quantum Speech Application Toolkit.arXiv preprint arXiv:2205.13221, 2022.
• (41)Robin Lorenz, Anna Pearson, Konstantinos Meichanetzidis, Dimitri Kartsaklis,and Bob Coecke.QNLP in Practice: Running Compositional Models of Meaning on aQuantum Computer.arXiv preprint arXiv:2102.12846, 2021.
• (42)Bob Coecke, Giovannide Felice, Konstantinos Meichanetzidis, and Alexis Toumi.Foundations for Near-Term Quantum Natural Language Processing.arXiv preprint arXiv:2012.03755, 2020.
• (43)Konstantinos Meichanetzidis, Alexis Toumi, Giovannide Felice, and Bob Coecke.Grammar-Aware Question-Answering on Quantum Computers.arXiv preprint arXiv:2012.03756, 2020.
• (44)Oleksandr Kyriienko, AnnieE Paine, and VincentE Elfving.Solving nonlinear differential equations with differentiable quantumcircuits.Physical Review A, 103(5):052416, 2021.
• (45)AnnieE. Paine, VincentE. Elfving, and Oleksandr Kyriienko.Physics-Informed Quantum Machine Learning: Solvingnonlinear differential equations in latent spaces without costly gridevaluations.arXiv preprint arXiv: 2308.01827, 2023.
• (46)AnnieE. Paine, VincentE. Elfving, and Oleksandr Kyriienko.Quantum kernel methods for solving regression problems anddifferential equations.Physical Review A, 107:032428, 2023.
• (47)MoKordzanganeh, Pavel Sekatski, Leonid Fedichkin, and Alexey Melnikov.An exponentially-growing family of universal quantum circuits.Machine Learning: Science and Technology, 4(3):035036, 2023.
• (48)Maria Schuld, Ryan Sweke, and JohannesJakob Meyer.Effect of data encoding on the expressive power of variationalquantum-machine-learning models.Physical Review A, 103(3):032430, 2021.
• (49)Maria Schuld, Alex Bocharov, KrystaM Svore, and Nathan Wiebe.Circuit-centric quantum classifiers.Physical Review A, 101(3):032308, 2020.
• (50)DavidE Rumelhart, GeoffreyE Hinton, and RonaldJ Williams.Learning representations by back-propagating errors.Nature, 323(6088):533–536, 1986.
• (51)Sebastian Ruder.An overview of gradient descent optimization algorithms.arXiv preprint arXiv: 1609.04747, 2016.
• (52)AtilimGunes Baydin, BarakA. Pearlmutter, AlexeyAndreyevich Radul, andJeffreyMark Siskind.Automatic differentiation in machine learning: a survey.Journal of Machine Learning Research, 18(153):1–43, 2018.
• (53)Kurt Hornik, MaxwellB. Stinchcombe, and HalbertL. White.Multilayer feedforward networks are universal approximators.Neural Networks, 2:359–366, 1989.
• (54)CGreenshields and HWeller.Notes on computational fluid dynamics: General principles.CFD Direct Ltd.: Reading, UK, 2022.
• (55)ParaView.https://www.paraview.org/, 2022.
• (56)DiederikP Kingma and Jimmy Ba.Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014.
• (57)Stefan Elfwing, Eiji Uchibe, and Kenji Doya.Sigmoid-weighted linear units for neural network functionapproximation in reinforcement learning.Neural Networks, 107:3–11, 2018.
• (58)Andrea Mari, ThomasR. Bromley, Josh Izaac, Maria Schuld, and Nathan Killoran.Transfer learning in hybrid classical-quantum neural networks.Quantum, 4:340, 2020.
• (59)John Preskill.Quantum computing in the nisq era and beyond.Quantum, 2:79, 2018.
• (60)Chen Zhao and Xiao-Shan Gao.QDNN: DNN with quantum neural network layers.arXiv preprint arXiv:1912.12660, 2019.
• (61)Tong Dou, Kaiwei Wang, Zhenwei Zhou, Shilu Yan, and Wei Cui.An unsupervised feature learning for quantum-classical convolutionalnetwork with applications to fault detection.In 2021 40th Chinese Control Conference (CCC), pages6351–6355. IEEE, 2021.
• (62)MichaelA Nielsen and Isaac Chuang.Quantum computation and quantum information, 2002.
• (63)QMware.https://qm-ware.com/, 2022.
• (64)Mohammad Kordzanganeh, Markus Buchberger, Basil Kyriacou, Maxim Povolotskii,Wilhelm Fischer, Andrii Kurkin, Wilfrid Somogyi, Asel Sagingalieva, MarkusPflitsch, and Alexey Melnikov.Benchmarking simulated and physical quantum processing units usingquantum and hybrid algorithms.Advanced Quantum Technologies, 6(8):2300043, 2023.
• (65)Tyson Jones and Julien Gacon.Efficient calculation of gradients in classical simulations ofvariational quantum algorithms.arXiv preprint arXiv:2009.02823, 2020.
• (66)Nikita Gourianov, Michael Lubasch, Sergey Dolgov, QuincyY vanden Berg, HessamBabaee, Peyman Givi, Martin Kiffner, and Dieter Jaksch.A quantum-inspired approach to exploit turbulence structures.Nature Computational Science, 2(1):30–37, 2022.
• (67)Michael Lubasch, Jaewoo Joo, Pierre Moinier, Martin Kiffner, and Dieter Jaksch.Variational quantum algorithms for nonlinear problems.Physical Review A, 101(1):010301, 2020.
• (68)Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, KaushikBhattacharya, etal.Neural operator: Learning maps between function spaces.arXiv preprint arXiv:2108.08481, 2021.
• (69)Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, KaushikBhattacharya, etal.Fourier neural operator for parametric partial differentialequations.arXiv preprint arXiv:2010.08895, 2020.
• (70)Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzalez, and PeterW. Battaglia.Learning mesh-based simulation with graph networks.In International Conference on Learning Representations, 2021.
• (71)Alvaro Sanchez-Gonzalez, Jonathan Godwin, Tobias Pfaff, Rex Ying, JureLeskovec, and Peter Battaglia.Learning to simulate complex physics with graph networks.In International Conference on Machine Learning, pages8459–8468. PMLR, 2020.