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arXiv:2301.07528 · IEEE International Geoscience and Remote Sensing Symposium (IGARSS)

Quantum-Inspired Tensor Network for Earth Science

Soronzonbold Otgonbaatar, Dieter August Kranzlmüller

Published Jul 16, 2023• 7 citations
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Abstract

Deep Learning (DL) is one of many successful methodologies to extract informative patterns and insights from ever increasing noisy large-scale datasets (in our case, satellite images). However, DL models consist of a few thousand to millions of training parameters, and these training parameters require tremendous amount of electrical power for extracting informative patterns from noisy large-scale datasets (e.g., computationally expensive). Hence, we employ a quantum-inspired tensor network for compressing trainable parameters of physics-informed neural networks (PINNs) in Earth science. PINNs are DL models penalized by enforcing the law of physics; in particular, the law of physics is embedded in DL models. In addition, we apply tensor decomposition to HyperSpectral Images (HSIs) to improve their spectral resolution. A quantum-inspired tensor network is also the native formulation to efficiently represent and train quantum machine learning models on big datasets on GPU tensor cores. Furthermore, the key contribution of this paper is twofold: (I) we reduced a number of trainable parameters of PINNs by using a quantum-inspired tensor network, and (II) we improved the spectral resolution of remotely-sensed images by employing tensor decomposition. As a benchmark PDE, we solved Burger’s equation. As practical satellite data, we employed HSIs of Indian Pine, USA and of Pavia University, Italy.

DOI
10.1109/igarss52108.2023.10282577

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Deep Learning (DL) is one of many successful methodologies to extract informative patterns and insights from ever increasing noisy large-scale datasets (in our case, satellite images). However, DL models consist of a few thousand to millions of training parameters, and these training parameters require tremendous amount of electrical power for extracting informative patterns from noisy large-scale datasets (e.g., computationally expensive). Hence, we employ a quantum-inspired tensor network for compressing trainable parameters of physics-informed neural networks (PINNs) in Earth science. PINNs are DL models penalized by enforcing the law of physics; in particular, the law of physics is embedded in DL models. In addition, we apply tensor decomposition to HyperSpectral Images (HSIs) to improve their spectral resolution. A quantum-inspired tensor network is also the native formulation to efficiently represent and train quantum machine learning models on big datasets on GPU tensor cores. Furthermore, the key contribution of this paper is twofold: (I) we reduced a number of trainable parameters of PINNs by using a quantum-inspired tensor network, and (II) we improved the spectral resolution of remotely-sensed images by employing tensor decomposition. As a benchmark PDE, we solved Burger’s equation. As practical satellite data, we employed HSIs of Indian Pine, USA and of Pavia University, Italy.
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Abstract Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large number of parameters to achieve reasonable accuracy, particularly for complex PDEs. In this paper, we present a quantum-classical PINNs (QCPINNs) that combines quantum and classical components, allowing us to solve PDEs with significantly fewer parameters while maintaining comparable accuracy and convergence to classical PINNs. We systematically evaluated two quantum circuit architectures across various configurations on five benchmark PDEs to identify optimal QCPINN designs. Our results demonstrate that the QCPINN achieves stable convergence and comparable accuracy, while using only 10%–30% of the trainable parameters required by classical PINNs. This approach also results in a significant reduction in the relative L 2 error for Helmholtz, Klein–Gordon, and Convection–diffusion equations, with reduction ranging from 4% to 64% across various fields. These findings demonstrate the potential of parameter efficiency and solution accuracy in physics-informed machine learning, allowing for a substantial decrease in model complexity without compromising solution quality.
Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding governing physical laws directly into the training objective. Recent advances in quantum machine learning have motivated hybrid quantum-classical extensions aimed at enhancing representational capacity while remaining compatible with near-term quantum hardware. In this work, we investigate trainable embedding strategies within quantum-assisted PINNs for solving parabolic PDEs, using one- and two-dimensional heat equations as canonical benchmarks. We introduce two quantum-assisted architectures that differ in their embedding components. In the first approach, a classical feed-forward neural network generates trainable feature maps for quantum data encoding (FNN-TE-QPINN). In the second, the embedding stage is realized entirely by a parameterized quantum circuit (QNN-TE-QPINN), yielding a fully quantum feature map. Our findings emphasize the critical role of embedding design and support hybrid quantum-classical approaches for parabolic PDE modeling in the NISQ era.
Recently, a novel type of Neural Network (NNs): the Physics-Informed Neural Networks (PINNs), was discovered to have many applications in computational physics. By integrating knowledge of physical laws and processes in Partial Differential Equations (PDEs), fast convergence and effective solutions are obtained. Since training modern Machine Learning (ML) systems is a computationally intensive endeavour, using Quantum Computing (QC) in the ML pipeline attracts increasing interest. Indeed, since several Quantum Machine Learning (QML) algorithms have already been implemented on present-day noisy intermediate-scale quantum devices, experts expect that ML on reliable, large-scale quantum computers will soon become a reality. However, after potential benefits from quantum speedup, QML may also entail reliability, trustworthiness, safety, and security risks. To solve the challenges of QML, we combine classical information processing, quantum manipulation, and processing with PINNs to accomplish a hybrid QML model named quantum based PINNs.
Remote sensing imagery classification remains a formidable challenge, primarily attributed to the high dimensionality and complexity inherent in the data. While deep learning has demonstrated efficacy in this domain, its computational demands and reliance on extensive training data pose significant hurdles. This research investigates how quantum computing could be able to overcome these constraints and improve deep learning’s capacity for classifying remote sensing data. The study provides an extensive review of recent developments in quantum deep learning techniques specifically tailored for remote sensing applications. Various quantum algorithms designed for image feature extraction and classification are discussed, shedding light on their potential to revolutionize the field. As a practical demonstration, the chapter presents a case study focused on leveraging quantum deep learning for land cover classification using Sentinel-2 imagery. This chapter’s main focus is on hybrid quantum convolutional neural networks (QCNNs) based on circuits as effective tools for remote sensing picture classification. These hybrid models integrate quantum layers within traditional neural networks, introducing a novel paradigm for image analysis. To evaluate the efficacy of this approach, the proposed QCNN architecture is applied to land-use and land-cover classification, employing the Euro SAT dataset as a benchmark.
This paper proposes a novel approach for solving nonlinear partial differential equations (PDEs) with a quantum computer, the trainable embedding quantum physics informed neural network (TE-QPINN). We combine quantum machine learning (QML) with physics informed neural networks (PINNs) in a hybrid approach. By leveraging the advantages of classical and quantum computers, we can create algorithms that have a potential to be run on noisy intermediate-scale quantum devices (NISQ). We use feedforward neural networks (FNN) as problem-agnostic embedding functions, giving the used quantum circuit greater expressibility than previously introduced embedding. This expressibility allows us to solve a wide range of problems without using a problem specific ansatz. Additionally, we introduce a hybrid backpropagation algorithm that allows efficient updates of the used weights and biases in the FNN embedding functions. In this paper we showcase the capabilities of TE-QPINNs of a wide range of problems, including the two-dimensional Poisson, Burgers and Navier-Stokes equations. In direct comparison with classical PINNs, this approach showed an ability to achieve superior results while using the same number of parameters, highlighting their potential for more efficient optimization in high-dimensional parameter spaces, which could be transformative for future applications.