Tensor Contraction Research Articles

Optical properties of quasi-two-dimensional objects from time-dependent density functional theory: Longitudinal versus transverse dielectric functions

Comprehension of the electronic properties of nano-objects is a key to defining dedicated properties, which can be adjusted by changing their size. Beyond confinement effects, the presence of interfaces, i.e., places where there is an abrupt change of electronic density, should also play a role. Time-dependent density functional theory (TD-DFT) is a state-of-the-art ab initio formalism in which this effect is accounted for through the so-called local field effects. In an earlier paper [S. Mazzei and C. Giorgetti, Phys. Rev. B 106, 035431 (2022)], we showed that the framework inherited from three-dimensional crystals could not provide reliable absorption spectra. In the present paper, we propose to calculate the macroscopic average of the dielectric tensor of a quasi-two-dimensional (2D) object from the response function of the density to the total macroscopic potential in order to avoid use of the so-called Adler and Wiser formula. We evidence that the inclusion of interfaces in the thickness of the slab causes the response function for the out-of-plane component to move sharply from the bulk absorption resonance to the plasmon one. This shows that the longitudinal-longitudinal contraction of the dielectric tensor is no longer equal to the transverse-transverse one in a quasi-2D object for out-of-plane perturbation. Nevertheless, we also show that the macroscopic average of the dielectric tensor of an ultrathin slab calculated within the longitudinal formalism of TD-DFT depicts the properties of the transverse reflectance and transmittance spectra of a thin slab.

Massively parallel GPU enabled third-order cluster perturbation excitation energies for cost-effective large scale excitation energy calculations.

We present here a massively parallel implementation of the recently developed CPS(D-3) excitation energy model that is based on cluster perturbation theory. The new algorithm extends the one developed in Baudin et al. [J. Chem. Phys., 150, 134110 (2019)] to leverage multiple nodes and utilize graphical processing units for the acceleration of heavy tensor contractions. Furthermore, we show that the extended algorithm scales efficiently with increasing amounts of computational resources and that the developed code enables CPS(D-3) excitation energy calculations on large molecular systems with a low time-to-solution. More specifically, calculations on systems with over 100 atoms and 1000 basis functions are possible in a few hours of wall clock time. This establishes CPS(D-3) excitation energies as a computationally efficient alternative to those obtained from the coupled-cluster singles and doubles model.

Real-Time Equation-of-Motion Coupled-Cluster Cumulant Green's Function Method: Heterogeneous Parallel Implementation Based on the Tensor Algebra for Many-Body Methods Infrastructure.

We report the implementation of the real-time equation-of-motion coupled-cluster (RT-EOM-CC) cumulant Green's function method [ J. Chem. Phys. 2020, 152, 174113] within the Tensor Algebra for Many-body Methods (TAMM) infrastructure. TAMM is a massively parallel heterogeneous tensor library designed for utilizing forthcoming exascale computing resources. The two-body electron repulsion matrix elements are Cholesky-decomposed, and we imposed spin-explicit forms of the various operators when evaluating the tensor contractions. Unlike our previous real algebra Tensor Contraction Engine (TCE) implementation, the TAMM implementation supports fully complex algebra. The RT-EOM-CC singles (S) and doubles (D) time-dependent amplitudes are propagated using a first-order Adams-Moulton method. This new implementation shows excellent scalability tested up to 500 GPUs using the Zn-porphyrin molecule with 655 basis functions, with parallel efficiencies above 90% up to 400 GPUs. The TAMM RT-EOM-CCSD was used to study core photoemission spectra in the formaldehyde and ethyl trifluoroacetate (ESCA) molecules. Simulations of the latter involve as many as 71 occupied and 649 virtual orbitals. The relative quasiparticle ionization energies and overall spectral functions agree well with available experimental results.

Toward an efficient implementation of internally contracted coupled-cluster methods.

A new implementation of the internally contracted multireference coupled-cluster with singles and doubles (icMRCCSD) method is presented. The new code employs an efficient tensor contraction kernel and can also avoid full four-external integral transformations, which significantly extends the scope of the applicability of icMRCCSD. The new implementation is currently restricted to the simple case of two active electrons in two orbitals and also supports the computation of spin-adapted doublet and triplet coupled-cluster wavefunctions. This contribution describes the basic approach for the automated derivation of working equations and benchmarks the current code against efficient implementations of standard methods, such as single-reference coupled-cluster singles and doubles (CCSD) and internally contracted multireference configuration interaction (icMRCI). Run times for linearized variants of icMRCCSD are only twice as long as comparable CCSD runs and similar to those of the icMRCI implementation, while non-linear terms of more complete variants of icMRCCSD lead to an order of magnitude longer computation times. Nevertheless, the new code allows for computations at larger scales than it was possible previously, with less demands on memory and disk-space resources. This is exemplified by numerical structure optimizations and harmonic force field determinations of NC2H5 isomers and the singlet and triplet states of m-benzyne. In addition, the exchange coupling of a dinuclear copper complex is determined. This work also defines a new commutator approximation for icMRCCSD, which includes all terms that are also present in the single-reference CCSD method, thus yielding a consistent pair of single-reference and multireference coupled-cluster methods.

Object Localization by Construction of an Asymmetric Isobody of the Magnetic Gradient Tensor Contraction Using Two Identical Permanent Magnets

This paper deals with predictive maintenance of infrastructure objects, focusing on the inspection of bridges. To achieve this goal, unmanned aerial vehicles place measuring units for data collection at different points of the bridge. This work deals with the question of how these measuring units can be found and removed by them later. For this purpose, a magnetic localization method based on the STAR method is presented. In contrast to the STAR method, which only detects the center of a magnetic target, a modified version called SRIOD is implemented here, where the orientation of the magnetic target can also be detected. This is essential information if the unmanned aerial vehicle is to grab the measuring unit with a robotic arm. In this regard, two identical permanent magnets are integrated into the measuring unit, whose gradient tensor contraction of the magnetic flux density distribution has a directed asymmetric surface when considering the contours of constant values. This creates a magnetic invisible reference plane that can be detected with a magnetic gradiometer. The method is verified both in simulation and experimentally in a laboratory setup.

Automatic transformation of irreducible representations for efficient contraction of tensors with cyclic group symmetry

Tensor contractions are ubiquitous in computational chemistry and physics, where tensors generally represent states or operators and contractions express the algebra of these quantities. In this context, the states and operators often preserve physical conservation laws, which are manifested as group symmetries in the tensors. These group symmetries imply that each tensor has block sparsity and can be stored in a reduced form. For nontrivial contractions, the memory footprint and cost are lowered, respectively, by a linear and a quadratic factor in the number of symmetry sectors. State-of-the-art tensor contraction software libraries exploit this opportunity by iterating over blocks or using general block-sparse tensor representations. Both approaches entail overhead in performance and code complexity. With intuition aided by tensor diagrams, we present a technique, irreducible representation alignment, which enables efficient handling of Abelian group symmetries via only dense tensors, by using contraction-specific reduced forms. This technique yields a general algorithm for arbitrary group symmetric contractions, which we implement in Python and apply to a variety of representative contractions from quantum chemistry and tensor network methods. As a consequence of relying on only dense tensor contractions, we can easily make use of efficient batched matrix multiplication via Intel’s MKL and distributed tensor contraction via the Cyclops library, achieving good efficiency and parallel scalability on up to 4096 Knights Landing cores of a supercomputer.

Codebase release 1.3 for SymTensor

Tensor contractions are ubiquitous in computational chemistry and physics, where tensors generally represent states or operators and contractions express the algebra of these quantities. In this context, the states and operators often preserve physical conservation laws, which are manifested as group symmetries in the tensors. These group symmetries imply that each tensor has block sparsity and can be stored in a reduced form. For nontrivial contractions, the memory footprint and cost are lowered, respectively, by a linear and a quadratic factor in the number of symmetry sectors. State-of-the-art tensor contraction software libraries exploit this opportunity by iterating over blocks or using general block-sparse tensor representations. Both approaches entail overhead in performance and code complexity. With intuition aided by tensor diagrams, we present a technique, irreducible representation alignment, which enables efficient handling of Abelian group symmetries via only dense tensors, by using contraction-specific reduced forms. This technique yields a general algorithm for arbitrary group symmetric contractions, which we implement in Python and apply to a variety of representative contractions from quantum chemistry and tensor network methods. As a consequence of relying on only dense tensor contractions, we can easily make use of efficient batched matrix multiplication via Intel’s MKL and distributed tensor contraction via the Cyclops library, achieving good efficiency and parallel scalability on up to 4096 Knights Landing cores of a supercomputer.

High Performance Hierarchical Tucker Tensor Learning Using GPU Tensor Cores

Extracting information from large-scale high-dimensional data is a fundamentally important task in high performance computing, where the hierarchical Tucker (HT) tensor learning approach (learning a tensor-tree structure) has been widely used in many applications. However, HT tensor learning algorithms are compute-intensive due to the “ <i>curse of dimensionality</i> ,” i.e., the time complexity grows exponentially with the order of the data tensor. The computation of HT tensor learning algorithms boils down to tensor primitives, which are amenable to computing on GPU tensor cores. Existing work does not support HT tensor learning using GPU tensor cores. There are three main challenges to address: 1) to accelerate tensor learning primitives using GPU tensor cores; 2) to implement the tensor learning algorithms using GPU tensor cores and multiple GPUs; 3) to support large-scale data tensors exceeding the GPU memory capacity. In this paper, we present efficient HT tensor learning primitives using GPU tensor cores and demonstrate three applications. First, we utilize GPU tensor cores to optimize HT tensor learning primitives, including tensor contractions, tensor matricizations and tensor singular value decomposition (SVD). We employ the optimized primitives to optimize HT tensor decomposition algorithms for Big Data analysis. Second, we propose a novel HT tensor layer for deep neural networks, whose training process only involves a forward pass without back propagation. The forward pass consists of tensor operations, thus further exploiting the computing power of GPU tensor cores. Third, we apply the optimized primitives to develop a tensor-tree structured quantum machine learning algorithm <i>tree-tensor network (TTN)</i> . Compared with TensorLy and TensorNetwork on NVIDIA A100 GPUs, our third-order HT tensor decomposition algorithm achieves up to <inline-formula><tex-math notation="LaTeX">$8.92 \times$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$6.42 \times$</tex-math></inline-formula> speedups, respectively, and our high-order case achieves up to <inline-formula><tex-math notation="LaTeX">$32.67 \times$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$23.97 \times$</tex-math></inline-formula> speedups, respectively. Our HT tensor layer for a fully connected neural network achieves <inline-formula><tex-math notation="LaTeX">$49.2 \times$</tex-math></inline-formula> compression at the cost of 0.5% drops in accuracy and <inline-formula><tex-math notation="LaTeX">$1.42 \times$</tex-math></inline-formula> speedup compared with the implementation on CUDA cores; for the AlexNet, our HT tensor layer achieves <inline-formula><tex-math notation="LaTeX">$9.45 \times$</tex-math></inline-formula> compression at the cost of 0.8% drops in accuracy and <inline-formula><tex-math notation="LaTeX">$1.87 \times$</tex-math></inline-formula> speedup compared with the implementation on CUDA cores. Our TTN algorithm achieves up to <inline-formula><tex-math notation="LaTeX">$11.17\times$</tex-math></inline-formula> speedup compared with TensorNetwork, indicating the potential of optimized tensor learning primitives for the classical simulation of quantum machine learning algorithms.

Spectral asymptotics for contracted tensor ensembles

Spectral asymptotics for contracted tensor ensembles

Polyhedral Specification and Code Generation of Sparse Tensor Contraction with Co-iteration

This article presents a code generator for sparse tensor contraction computations. It leverages a mathematical representation of loop nest computations in the sparse polyhedral framework (SPF), which extends the polyhedral model to support non-affine computations, such as those that arise in sparse tensors. SPF is extended to perform layout specification, optimization, and code generation of sparse tensor code: (1) We develop a polyhedral layout specification that decouples iteration spaces for layout and computation; and (2) we develop efficient co-iteration of sparse tensors by combining polyhedra scanning over the layout of one sparse tensor with the synthesis of code to find corresponding elements in other tensors through an SMT solver. We compare the generated code with that produced by a state-of-the-art tensor compiler, TACO. We achieve on average 1.63× faster parallel performance than TACO on sparse-sparse co-iteration and describe how to improve that to 2.72× average speedup by switching the find algorithms. We also demonstrate that decoupling iteration spaces of layout and computation enables additional layout and computation combinations to be supported.

A Native Tensor–Vector Multiplication Algorithm for High Performance Computing

Tensor computations are important mathematical operations for applications that rely on multidimensional data. The tensor–vector multiplication (TVM) is the most memory-bound tensor contraction in this class of operations. This article proposes an open-source TVM algorithm which is much simpler and efficient than previous approaches, making it suitable for integration in the most popular BLAS libraries available today. Our algorithm has been written from scratch and features unit-stride memory accesses, cache awareness, mode obliviousness, full vectorization and multi-threading as well as NUMA awareness for non-hierarchically stored dense tensors. Numerical experiments are carried out on tensors up to order 10 and various compilers and hardware architectures equipped with traditional DDR and high bandwidth memory (HBM). For large tensors the average performance of the TVM ranges between 62% and 76% of the theoretical bandwidth for NUMA systems with DDR memory and remains independent of the contraction mode. On NUMA systems with HBM the TVM exhibits some mode dependency but manages to reach performance figures close to peak values. Finally, the higher-order power method is benchmarked with the proposed TVM kernel and delivers on average between 58% and 69% of the theoretical bandwidth for large tensors.

Neural tensor contractions and the expressive power of deep neural quantum states

We establish a direct connection between general tensor networks and deep feed-forward artificial neural networks. The core of our results is the construction of neural-network layers that efficiently perform tensor contractions and that use commonly adopted nonlinear activation functions. The resulting deep networks feature a number of edges that closely match the contraction complexity of the tensor networks to be approximated. In the context of many-body quantum states, this result establishes that neural-network states have strictly the same or higher expressive power than practically usable variational tensor networks. As an example, we show that all matrix product states can be efficiently written as neural-network states with a number of edges polynomial in the bond dimension and depth that is logarithmic in the system size. The opposite instead does not hold true, and our results imply that there exist quantum states that are not efficiently expressible in terms of matrix product states or projected entangled pair states but that are instead efficiently expressible with neural network states.

Using tensor contractions to derive the structure of slice-wise multiplications of tensors with applications to space–time Khatri–Rao coding for MIMO-OFDM systems

The slice-wise multiplication of two tensors is required in a variety of tensor decompositions (including PARAFAC2 and PARATUCK2) and is encountered in many applications, including the analysis of multidimensional biomedical data (EEG, MEG, etc.) or multi-carrier multiple-input multiple-output (MIMO) systems. In this paper, we propose a new tensor representation that is not based on a slice-wise (matrix) description, but can be represented by a double contraction of two tensors. Such a double contraction of two tensors can be efficiently calculated via generalized unfoldings. It leads to new tensor models of the investigated system that do not depend on the chosen unfolding (in contrast to matrix models) and reveal the tensor structure of the data model, such that all possible unfoldings can be seen at the same time. As an example, we apply this new concept to the design of new receivers for multi-carrier MIMO systems in wireless communications. In particular, we consider MIMO-orthogonal frequency division multiplexing (OFDM) systems with and without Khatri–Rao coding. The proposed receivers exploit the channel correlation between adjacent subcarriers, require the same amount of training symbols as traditional OFDM techniques, but have an improved performance in terms of the symbol error rate. Furthermore, we show that the spectral efficiency of the Khatri–Rao-coded MIMO-OFDM can be increased by introducing cross-coding such that the “coding matrix” also contains useful information symbols. Considering this transmission technique, we derive a tensor model and two types of receivers for cross-coded MIMO-OFDM systems using the double contraction of two tensors.

A tale of two butterflies: an exact equivalence in higher-derivative gravity

We prove the equivalence of two holographic computations of the butterfly velocity in higher-derivative theories with Lagrangian built from arbitrary contractions of curvature tensors. The butterfly velocity characterizes the speed at which local perturbations grow in chaotic many-body systems and can be extracted from the out-of-time-order correlator. This leads to a holographic computation in which the butterfly velocity is determined from a localized shockwave on the horizon of a dual black hole. A second holographic computation uses entanglement wedge reconstruction to define a notion of operator size and determines the butterfly velocity from certain extremal surfaces. By direct computation, we show that these two butterfly velocities match precisely in the aforementioned class of gravitational theories. We also present evidence showing that this equivalence holds in all gravitational theories. Along the way, we prove a number of general results on shockwave spacetimes.

Accelerating Tensor Contraction Products via Tensor-Train Decomposition [Tips &amp; Tricks

Tensors (multiway arrays) and tensor decompositions (TDs) have recently received tremendous attention in the data analytics community, due to their ability to mitigate the curse of dimensionality associated with modern large-dimensional big data <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> . Indeed, TDs allow for data volume (e.g., the parameter complexity) to be reduced from scaling exponentially to scaling linearly in the tensor dimensions, which facilitates applications in areas including the compression and interpretability of neural networks <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> , multimodal learning <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , and completion of knowledge graphs <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[4]</xref> , <xref ref-type="bibr" rid="ref5" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[5]</xref> . At the heart of TD techniques is the tensor contraction product (TCP), an operator used for representing even the most unmanageable higher-order tensors through a set of small-scale core tensors that are interconnected via TCP operations <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> .

Distributed-memory tensor completion for generalized loss functions in python using new sparse tensor kernels

Tensor computations are increasingly prevalent numerical techniques in data science, but pose unique challenges for high-performance implementation. We provide novel algorithms and systems infrastructure which enable efficient parallel implementation of algorithms for tensor completion with generalized loss functions. Specifically, we consider alternating minimization, coordinate minimization, and a quasi-Newton (generalized Gauss-Newton) method. By extending the Cyclops library, we implement all of these methods in high-level Python syntax. To make possible tensor completion for very sparse tensors, we introduce new multi-tensor primitives, for which we provide specialized parallel implementations. We compare these routines to pairwise contraction of sparse tensors by reduction to hypersparse matrix formats, and find that the multi-tensor routines are more efficient in theoretical cost and execution time in experiments. We provide microbenchmarking results on the Stampede2 supercomputer to demonstrate the efficiency of the new primitives and Cyclops functionality. We then study the performance of the tensor completion methods for a synthetic tensor with 10 billion nonzeros and the Netflix dataset, considering both least squares and Poisson loss functions.

Quantum Perturbation Theory Using Tensor Cores and a Deep Neural Network.

Time-independent quantum response calculations are performed using Tensor cores. This is achieved by mapping density matrix perturbation theory onto the computational structure of a deep neural network. The main computational cost of each deep layer is dominated by tensor contractions, i.e., dense matrix-matrix multiplications, in mixed-precision arithmetics, which achieves close to peak performance. Quantum response calculations are demonstrated and analyzed using self-consistent charge density-functional tight-binding theory as well as coupled-perturbed Hartree-Fock theory. For linear response calculations, a novel parameter-free convergence criterion is presented that is well-suited for numerically noisy low-precision floating point operations and we demonstrate a peak performance of almost 200 Tflops using the Tensor cores of two Nvidia A100 GPUs.

New discovery and short-term effect analysis of tensor tympani muscle tenotomy for Meniere's disease under otoscope

Objective: To explore the feasibility and short-term effect of tensor tympani muscle Tenotomy in the treatment of Meniere's disease under otoscope. The possible pathogenesis was discussed and our views were put forward. Methods: The clinical data of 9 cases of Meniere's disease treated by otoscopic Tenotomy were analyzed retrospectively, including 2 males, 7 females, 5 right ones, 2 left ones and 2 bilateral ones. The average age was (56.33± 10.56) years, ranging from 38 to 75 years. We evaluated intraoperative findings and short-term postoperative efficacy, respectively evaluated postoperative aural fullness, tinnitus and hearing recovery, and evaluated postoperative vertigo attack in a short time. Results: Nine patients were completed the operation under general anaesthesia and otoscopy, and no serious complications occurred. We found new pathological changes in tympanic cavity in some cases during operation. There were rupture of round window membrane in 1 case, severe fibrous hyperplasia near the round window membrane and vestibular window and adhesion with ossicular chain in 1 case, fibrous cord and membranous hyperplasia near vestibular window and round window membrane in 1 case, fibrous hyperplasia and adhesion near the round window membrane in 2 cases, membranous hyperplasia and adhesion around vestibular window in 1 case. No fibrous hyperplasia was found in 3 cases in the tympanic cavity. The round window membrane can be exposed in 4 cases and failed in 5 cases. After 3 months of follow-up, we found that we found that 5/5 cases of aural fullness disappeared, 2/2 cases of earache disappeared, 3/8 cases of tinnitus improved, 5/8 cases presented with improvement and no aggravation, 3/3 cases of hearing allergy improved, 4/9 cases of hearing improved, and 5/9 cases showed no improvement or decrease. 9 patients were followed up for 3 months, of whom 8 patients had no vertigo, one patient suffered from vertigo twice within 3 months after operation, and the patient suffered from rupture of round window membrane. Conclusions: Endoscopic Tenotomy for Meniere's disease has obvious curative effect and quick recovery after operation. During the operation, we find that most of Meniere's patients have fibrous cord hyperplasia near the inner ear window membrane, which may be the pathological manifestation after repeated rupture and repair of the inner ear window membrane. The vertigo of Meniere's disease may be related to the destruction and repair of inner ear membrane structure caused by improper contraction or spasm of tympanic tensor muscle.

Noise-Induced Entanglement Transition in One-Dimensional Random Quantum Circuits

A random quantum circuit is a minimally structured model to study entanglement dynamics of many-body quantum systems. We consider a one-dimensional quantum circuit with noisy Haar-random unitary gates using density matrix operator and tensor contraction methods. It is shown that the entanglement evolution of the random quantum circuits is properly characterized by the logarithmic entanglement negativity. By performing exact numerical calculations, we find that, as the physical error rate is decreased below a critical value p c ≈ 0.056, the logarithmic entanglement negativity changes from the area law to the volume law, giving rise to an entanglement transition. The critical exponent of the correlation length can be determined from the finite-size scaling analysis, revealing the universal dynamic property of the noisy intermediate-scale quantum devices.

Functional tensor network solving many-body Schrödinger equation

Schr\"odinger equation belongs to the most fundamental differential equations in quantum physics. However, the exact solutions are extremely rare, and many analytical methods are applicable only to the cases with small perturbations or weak correlations. Solving the many-body Schr\"odinger equation in the continuous spaces with the presence of strong correlations is an extremely important and challenging issue. In this work, we propose the functional tensor network (FTN) approach to solve the many-body Schr\"odinger equation. Provided the orthonormal functional bases, we represent the coefficients of the many-body wave-function as tensor network. The observables, such as energy, can be calculated simply by tensor contractions. Simulating the ground state becomes solving a minimization problem defined by the tensor network. An efficient gradient-decent algorithm based on the automatically differentiable tensors is proposed. We here take matrix product state (MPS) as an example, whose complexity scales only linearly with the system size. We apply our approach to solve the ground state of coupled harmonic oscillators, and achieve high accuracy by comparing with the exact solutions. Reliable results are also given with the presence of three-body interactions, where the system cannot be decoupled to isolated oscillators. Our approach is simple and with well-controlled error, superior to the highly-nonlinear neural-network solvers. Our work extends the applications of tensor network from quantum lattice models to the systems in the continuous space. FTN can be used as a general solver of the differential equations with many variables. The MPS exemplified here can be generalized to, e.g., the fermionic tensor networks, to solve the electronic Schr\"odinger equation.