Light beam transmission characteristic model discovery and reconstruction method based on sparse physical information neural network

Abstract

The invention discloses a physical model discovery and reconstruction method for light beam transmission characteristics based on a sparse physical information neural network, and belongs to the technical field of nonlinear optics and deep learning intersection, and the method comprises the following steps: S1, making a sparse and low-fidelity data set; s2, establishing a candidate function library of items possibly existing in a data back model by using automatic differentiation; S3, constructing a physical information neural network model with sparse regression to discover a physical model structure; s4, coefficient fine tuning is carried out on the found physical model, and high-fidelity data is reconstructed.The method is suitable for model discovery of light beam transmission data supported by a constant coefficient or variable coefficient partial differential equation, scarce and noisy low-fidelity data can be utilized, and high-fidelity data can be obtained. A simple physical model is found according to a physical information neural network and sparse regression, and a light beam transmission dynamic process is reconstructed, so that subsequent theoretical analysis and research on light beam transmission characteristics are facilitated.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of the cross of nonlinear optics and deep learning, more particularly, it is particularly related to the reconstruction of supported physical model equation and high-fidelity solution from the light beam transmission process controlled by nonlinear Schrodinger type equation. BACKGROUND

[0002] Nonlinear partial differential equations are the core tool for describing the spatiotemporal evolution of optical fields, especially in the field of light beam transmission in complex media. When high-intensity laser propagates in optical fibers or nonlinear crystals, its dynamic behavior is dominated by the complex interaction of dispersion effect, nonlinear effect and diffraction effect, and these processes are usually modeled by generalized nonlinear Schrodinger equation.

[0003] In the case where the PDE display expression is known, the modeling analysis is usually carried out by using traditional numerical methods such as Newton iteration method, step-by-step Fourier method, fourth-order Runge-Kutta method, etc. However, the modeling cost of numerical method is expensive and requires deep professional knowledge and mathematical theory. With the improvement of the availability of observation data and the popularity of artificial intelligence in scientific and engineering fields, data-driven machine learning provides a new way for the research of light beam transmission. In order to integrate some prior bottom physical knowledge into neural network learning, physical information neural network is proposed to study the forward / backward problem of partial differential equation under the condition that only a small amount of data is available.

[0004] However, when dealing with systems whose control equation is partially or completely unknown, it is still a difficulty to discover the physical control equation and reconstruct the spatiotemporal dynamics of light beam transmission from sparse sampling data. Especially for some nonlinear heterogeneous media, variable coefficient model often needs to be considered. However, the current research methods are mostly based on constant coefficient terms and real number solutions, and cannot well handle the discovery of variable coefficient and complex number solution model, therefore, it is necessary to provide an algorithm suitable for the discovery of variable coefficient and complex number system model. The present application realizes the intelligent discovery from a small amount of observation data to a complete physical model, and provides a new paradigm for the research of complex light beam transmission. SUMMARY

[0005] In view of the above shortcomings in the prior art, the main purpose of the present application is to provide a model discovery and reconstruction method for light beam transmission characteristics based on sparse physical information neural network, which aims to solve the existing technical problems.

[0006] The technical scheme of the present application is as follows:

[0007] S1, preparing a sparse and low-fidelity data set;

[0008] S2, using automatic differentiation to establish a candidate function library of possible terms behind the data;

[0009] S3, constructing a physical information neural network model with sparse regression to discover the physical model structure of the beam propagation characteristics;

[0010] S4, performing coefficient fine-tuning on the discovered beam propagation model and reconstructing a high-fidelity beam propagation process;

[0011] As preferred, the specific step S1 of making a sparse and low-fidelity data set comprises the following sub-steps:

[0012] S11, obtaining a three-dimensional data set with a spatial size of 256x256 and a time size of 201 by using the fourth-order Runge-Kutta method as high-precision data;

[0013] S12, simulating sparse data in a real environment by randomly sparse sampling 2500 spatial points and 15 time points (about 0.3% of the total data set), and taking them as the data set for neural network training.

[0014] As preferred, the data set in S1 is divided into training set and validation set according to the ratio of 8:2.

[0015] As preferred, an automatic differentiation is used to establish a candidate function library of possible terms behind the data. Assuming that the physical law behind the data is dominated by only a few important terms, these important terms can be selected from a large space library of candidate functions, and a candidate library function is constructed based on the possible terms (taking the discovery of nonlinear Schrödinger equation as an example), and the step S2 comprises the following sub-steps:

[0016] S21, considering that the transmission of the light beam is a multi-dimensional space-time system, its control equation can be described by a set of nonlinear, time-varying, parameterized partial differential equations, which has the general form:

[0017]

[0018] Where q=q(x,y,t) is the complex solution of the light beam transmission, composed of real part u and imaginary part v, Φ is the term that may exist behind the sparse observation data, and ξ is the coefficient before the term.

[0019] S22, constructing a virtual-real separated candidate library function according to physical knowledge, which is composed of solution terms {u, v}, factorial terms {u 2 ,v 2 ,u 3 ,v 3}, derivative terms {u x ,v x ,u y ,v y ,u xx ,v xx ,u yy ,v yy}, nonlinear terms {|q|2 u,|q| 2 v}, variable coefficient terms {exp(-t), cos(t)}, pump source terms {sin(t), cos(t), exp(-t),} and their combination together constitute the total candidate function library Φ

[0020] S23. Considering that the solution of the nonlinear Schrödinger equation is in complex form, we can separate the real and imaginary forms and obtain the following two forms:

[0021]

[0022] Where f u and f v are the real and imaginary parts after separation, u t and v t are the first-order derivatives of the real and imaginary solutions with respect to time, Φ is the candidate function library, ξ1 and ξ2 are coefficients

[0023] Preferably, a physical information neural network model with sparse regression is constructed to discover the physical model structure, and the step S3 includes the following sub-steps:

[0024] S31. Establish a suitable neural network model and hyperparameters, including network structure (number of network layers and number of neurons in each layer), learning rate, activation function, and optimizer;

[0025] S32. Construct the loss function of the physical information neural network, including two parts: the data part and the physical equation;

[0026] The loss function of the data constructed in the sub-step S32 is:

[0027]

[0028] Where N is the number of prior data points in S1, q i is the actual observed value, q θ is the predicted value output by the neural network;

[0029] The loss function of constructing the physical equation in the sub-step S32 is:

[0030]

[0031] Where M is the number of collocation points used to calculate the physical equations, and are the real and imaginary physical residual terms of formula (4) after the separation of real and imaginary in S23, U(θ) includes u(θ) and v(θ), ξ includes ξ1 and ξ2;

[0032] The total loss function constructed in the sub-step S32 is:

[0033]

[0034] In the formula, μ is a weight coefficient of a physical loss function, used to balance the data impact, where α = 10; γ is a regularization parameter, and ||·||0 is an l0 norm;

[0035] S33, a sparse regression method of open source alternating direction optimization is used to alternately train the hyperparameters θ and the coefficients ξ of the physical information neural network by continuously adjusting a threshold parameter, and the alternating optimization formula is as follows:

[0036]

[0037] S34, an optimizer is set, and an optimizer combined by Adam and L-BFGS is used to continuously update the parameters θ and the coefficients ξ in iteration; and the value of the loss function is minimized, so that a simple physical equation structure supported by sparse data is finally found.

[0038] As preferred, the physical model found in S3 is subjected to coefficient fine-tuning and reconstruction of high-fidelity data;

[0039]

[0040] In the formula, a1~c1 are coefficients of found real part terms, and a2~c2 are coefficients of found imaginary part terms

[0041] A complex fitting method is used to integrate the found real part and imaginary part equations into a complete complex solution equation, and the steps follow the following rules:

[0042] (1), the found real part terms and imaginary part terms should be corresponding terms (such as u xx corresponding to v xx );

[0043] (2), if the signs before the corresponding real part terms and imaginary part terms are opposite, there is an imaginary number i before the fitted term; if the signs are the same, there is no imaginary number i;

[0044] (3), the sign of the fitted term follows the real part;

[0045] (4), the coefficient of the fitted term should be the average of the real part and the imaginary part;

[0046]

[0047] (5), the pump source term is considered independently;

[0048] The present application has the following beneficial effects:

[0049] 1. The present application starts from sparse and low-fidelity light beam data transmission, discovers the physical equation behind the data expression of space-time dynamics through a physical information neural network with sparse regression, and reconstructs a high-precision prediction solution. It realizes the discovery of light beam transmission model from a small amount of observation data and the reconstruction of complete light beam transmission process, providing a new paradigm for complex light beam control.

[0050] 2. Compared with the current mainstream physical information neural network model, the present application mainly highlights the discovery of potential equations from data under unknown model conditions, providing a unique perspective for subsequent theoretical modeling research. The method proposed by the present application is suitable for reconstructing the dominant equation of light beam transmission with different physical correction terms, such as nonlinear Schrödinger equation with different correction terms.

[0051] 3. The technical solution of the present application can reconstruct a high-precision prediction solution to describe the characteristics of light beam transmission, providing an efficient method for subsequent analysis of more complex physical mechanisms using physical equations.

[0052] 4. Compared with the current mainstream research range of constant coefficient and real number solution equation, the present application provides an improved candidate library function construction idea containing variable coefficient terms, which can be applied to discover variable coefficient equations, and proposes a complex fitting method suitable for discovering different types of complex number solution equations. BRIEF DESCRIPTION OF DRAWINGS

[0053] The accompanying drawings, which form a part of this application, are included to provide a further understanding of the application, and are incorporated in and constitute a part of this application. The embodiments of the present application and their descriptions are used to explain the present application, and do not constitute an improper limitation on the present application. In the drawings:

[0054] Figure 1 is the workflow diagram in the present application, which is divided into three stages;

[0055] Figure 2 is the algorithm flowchart in the present application;

[0056] Figure 3 is Figure 1 the first stage in

[0057] Figure 4 is Figure 1 the second stage in

[0058] Figure 5is Figure 1 In the third stage, data post-processing; the discovered beam propagation model is fine-tuned in coefficients and the high-fidelity beam propagation process is reconstructed;

[0059] Figure 6 is the three-stage evolution process of the sparse coefficients of the 68 candidate functions involved in the examples of the present application; Figure 6 (a) and Figure 6 (b) are the real part and the imaginary part of the discovered (2+1) -dimensional nonlinear Schrödinger equation, respectively;

[0060] Figure 7 is the reconstructed high-fidelity variable-coefficient vortex beam propagation process. Figure 7 (a) and Figure 7 (b) are the reconstructed vortex beam prediction solution and the absolute error, respectively, Figure 7 (c) is the maximum amplitude of the reconstructed prediction solution and the true solution; DETAILED DESCRIPTION

[0061] In order to make the above objectives, features and advantages of the present application more clear and obvious, the specific embodiments of the present application are described in detail below with reference to the description and drawings.

[0062] Obviously, in the following description, many specific details are set forth in order to provide a thorough understanding of the purpose features and advantages of the present application, but the present application can also be implemented in other ways different from the description, and those skilled in the art can make similar explorations without departing from the connotation of the present application, therefore the present application is not limited to the specific embodiments disclosed below.

[0063] Secondly, the "one embodiment" or "embodiment" referred to herein means that the specific features, structures or characteristics can be included in at least one implementation of the present application. "In one embodiment" appearing in different places in the specification does not mean the same embodiment, nor is it an independent or alternative embodiment that excludes other embodiments.

[0064] The terms "first", "second", etc. serve as a description, and cannot be understood as measuring the relative weight or implicitly indicating other technical parameters. Therefore, the features described with "first", "second", etc. can include one or more of the features explicitly or implicitly.

[0065] The present application discloses a sparse-based physical information neural network model discovery and reconstruction method for beam propagation characteristics, comprising the following steps:

[0066] S1, making a sparse and low-fidelity data set;

[0067] S2, using automatic differentiation to establish a candidate function library of possible terms behind the data model;

[0068] S3, constructing a physical information neural network model with sparse regression to discover the structure of the physical model;

[0069] S4, fine-tuning the coefficients of the discovered physical model and reconstructing high-fidelity data

[0070] Embodiment 1

[0071] Construct 30 improved term candidate library functions, which include constant terms and variable coefficient terms, which are composed of the combination of the solution terms [u, v, u x ,|q| xx ,|q| y ,|q| yy ,|q| x ,|q| xx ,|q| y ,|q| yy ,|q| 2 ,|q| 2 ,|q| and the coefficient term [1, exp(-t), cos(t)] (here, the negative exponential function and the cosine function are taken as examples, but it is not limited to this) and the pumping term [cos(t), sin(t), exp(-t)];

[0072] Referring to the workflow diagram of Figure 1 and the algorithm flowchart of Figure 2 , a discovery of a whispering gallery mode microcavity model (2+1) dimensional Lugiato-Lefever equation is completed. Referring to Figure 3 , the steps in S1 are made, and a high-precision numerical solution of [256x256x201] is made by using the fourth-order Runge-Kutta method, wherein the initial solution is:

[0073]

[0074] The spatial accuracy x and y at each time is 256x256 and x∈(-25, 25), y∈(-25, 25), and the time accuracy t is 201 and t∈(0, 5);

[0075] Figure 4 The steps in S2-S3 are performed to discover the basic structure of the Lugiato-Lefever equation behind the data support:

[0076]

[0077] Figure 5 The step S4 is performed to fine-tune the coefficients of the discovered physical model:

[0078]

[0079] Finally, the final complex function equation model is obtained by the fitting method of formula (9):

[0080] q t =i0.505q xx +i0.5q yy +0.9665|q| 2 q+i0.5265q-0.202q+0.497cos(t) (13)

[0081] Example 2

[0082] 68 constant term candidate library functions are constructed by constant [1], solution [u, v], derivative [u x ,u xx ,u y ,u yy ,v x ,v xx ,v y ,v yy ], factorial [u 2 ,u 3 ,v 2 ,v 3 ], second-order nonlinear [|q| 2 u,|q| 2 v] and its linear combination, and pumping term [cos(t), sin(t), exp(-t)];

[0083] Using the fourth-order Runge-Kutta method, a high-precision numerical solution of [256x256x201] is made, wherein the initial solution is the same as formula (10);

[0084] The spatial accuracy x and y is 256x256 and x∈(-40, 40), y∈(-40, 40) at each time, and the time accuracy t is 201 and t∈(0, 2);

[0085] Reference Figure 6 Three complete processes for the coefficient changes of the 68 candidate library functions are performed to complete steps S1-S4 to describe the discovery of the (2+1) dimensional standard nonlinear Schrödinger equation under the joint action of self-focusing and Kerr nonlinearity:

[0086]

[0087] Finally, the final complex function equation model is obtained by the fitting method of formula (9):

[0088] q t =i0.4985q xx +i0.4995q yy +i0.9945|q| 2 q (15)

[0089] Example 3

[0090] The same 30 candidate library functions as in Example 1 are constructed;

[0091] The high-precision numerical solution of [256x256x201] is made by using the fourth-order Runge-Kutta method, wherein the initial solution is the same as formula (10);

[0092] The spatial accuracy of x and y is 256x256 and x∈(-25, 25), y∈(-25, 25) at each time, and the time accuracy t is 201 and t∈(0, 20);

[0093] Referring to Figure 7 (a) and Figure 7 (b) are the absolute error of the reconstructed vortex beam prediction solution and the numerical solution under the condition that the diffraction coefficient and the Kerr nonlinear coefficient are both 0.5cos(t), respectively. Figure 7 (a) indicates that the width of the beam evolves periodically with time. Figure 7 (b) indicates that the maximum absolute error of the prediction appears near the origin and does not exceed 0.07. Figure 7 (c) indicates that the maximum amplitude variation of the prediction solution presents periodicity and is highly consistent with the true value.

[0094] The above describes in detail a method for model discovery and reconstruction of beam transmission characteristics based on a sparse physical information neural network according to an embodiment of the present application. In this paper, specific examples are used to describe the phenomena and implementation modes of the present application in detail. The above description of the embodiments is only for the purpose of helping to understand the present application and its advantages; at the same time, for those skilled in the art, the scheme of the present application can be modified or replaced, and corresponding changes will occur in the actual operation and use field, which are all covered within the scope of the present application.

Claims

1. A method for model discovery and reconstruction of beam propagation characteristics based on sparse physical information neural networks, characterized in that: The steps include: S1. Create a sparse and low-fidelity dataset; S2. Use automatic differentiation to build a library of candidate functions that may exist in the model behind the data; S3. Construct a physical information neural network model with sparse regression to discover the physical model structure of the beam transmission characteristics; S4. Fine-tune the coefficients of the discovered beam propagation model and reconstruct the high-fidelity beam propagation process.

2. The method for discovering and reconstructing a model of light beam transmission characteristics based on a sparse physical information neural network according to claim 1, wherein: The data set preparation in step S1 includes the following sub-steps: S11. Using the fourth-order Runge-Kutta method, a three-dimensional data set with a spatial size of 256×256 and a temporal size of 201 is obtained as high-precision data; S12. Randomly sparsely sample 2500 spatial points and 15 time points (about 0.3% of the total data set) to simulate sparse data in a real environment and use them as the data set for neural network training.

3. The method for discovering and reconstructing a model of light beam transmission characteristics based on a sparse physical information neural network according to claim 1, wherein: Automatic differentiation is used to build a library of candidate functions that may exist in the model behind the data. Assuming that the physical laws behind the data are governed by only a few important terms, these important terms can be selected from a large library of candidate functions. Based on the possible terms, a candidate library function is constructed (taking the discovery of the nonlinear Schrödinger equation as an example). Step S2 includes the following sub-steps: S21. Considering that the propagation of a light beam is a multidimensional space-time system, its governing equations can be described by a set of nonlinear, time-varying, parameterized PDEs, the general form of which is: where q = q(x, y, t) is the complex solution of the beam propagation, consisting of the real part u and the imaginary part v, Φ is the term that may exist behind the sparse observation data, and ξ is the coefficient before the term; S22, based on physical knowledge, construct candidate library functions for virtual-real separation, consisting of solution terms {u,v}, factorial terms {u 2 ,v 2 ,u 3 ,v 3 }、derivative term {u x ,v x ,u y ,v y ,u xx ,v xx ,u yy ,v yy }, nonlinear term {|q| 2 u,|q| 2 v}, variable coefficient terms {exp(-t), cos(t)}, pump source terms {sin(t), cos(t), exp(-t),} and their combination together constitute the total candidate function library Φ; S23. Considering that the solution of the nonlinear Schrödinger equation is in complex form, we can separate the real and imaginary forms and obtain the following two forms: Where f u and f v are the real and imaginary parts after separation, u t and v t are the first-order derivatives of the real and imaginary solutions with respect to time, Φ is the candidate function library, and ξ1 and ξ2 are coefficients.

4. The method for discovering and reconstructing a model of light beam transmission characteristics based on a sparse physical information neural network according to claim 1, wherein: The step S3 comprises the following sub-steps: S31. Establish a suitable neural network model and hyperparameters, including network structure (number of network layers and number of neurons in each layer), learning rate, activation function, and optimizer; S32. Construct the total loss function of the physical information neural network, including two parts: the data part and the physical equation; The total loss function constructed in the sub-step S32 is: Where N is the number of prior data points in S1, M is the number of collocation points used for physical equation calculation; q i is the actual observed value, q θ is the predicted value output by the neural network; U(θ) includes u(θ) and v(θ), ξ includes ξ1 and ξ2; μ is the weight coefficient of the physical loss function, which is used to balance the influence of data; γ is the regularization parameter, and ||·||0 is the l0norm; S33. Using an open-source alternating direction optimization sparse regression method, by continuously adjusting the threshold parameters, the hyperparameters θ and coefficients ξ of the physical information neural network are alternately trained. The alternating optimization formula is as follows; S34. Set up the optimizer and use the optimizer combined with Adam and L-BFGS57 to continuously update the parameters θ and coefficients ξ in the iteration; to minimize the value of the loss function, and finally discover the simple physical equation structure supported by the sparse data.

5. The method for discovering and reconstructing a model of light beam transmission characteristics based on a sparse physical information neural network according to claim 4, wherein: Fine-tune the coefficients of the beam propagation model discovered by S3 and reconstruct high-fidelity data; Where a1~c1 are the coefficients of the real part found, and a2~c2 are the coefficients of the imaginary part found. The imaginary and real part equations found are integrated into a complete complex equation solution using a proposed complex fitting method. The steps described follow the following rules: (1) The real and imaginary terms found should be corresponding terms that exist in pairs (such as u xx Corresponding to v xx ); (2) If the signs of the corresponding real and imaginary terms are opposite, the imaginary number i will be in front of the fitted term; if the signs are the same, there will be no imaginary number i; (3) The sign of the fitted term follows the real part; (4) The coefficient of the fitted term should be the average of the real and imaginary parts; (5) The pump source term is considered independently; Take a standard (2+1)-dimensional nonlinear Schrödinger equation as an example: Find the real and imaginary governing equations from sparse data: The proposed complex fitting method is used to integrate the discovered imaginary and real part equations into a complete complex solution equation: q t =i0.4985q xx +i0.4995q yy +i0.9945|q| 2 q (9).

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