One-dimensional function fitting method and system based on input dimension expansion

Abstract

The invention relates to a one-dimensional function fitting method and system based on input dimension expansion, and the method comprises the steps: firstly obtaining a target function and a training sample thereof, carrying out the dimension expansion of each input scalar according to an odd-dimension central symmetry rule: placing the scalar at the center of an expansion vector, and filling the two sides with the same constant related to a domain of definition; and constructing a neural network taking the extended vector as input, and adjusting the width of the hidden layer to enable the deviation between the total parameter number and the reference model not to exceed a set value, thereby realizing parameter redistribution. Then, a second-order optimization algorithm is used to train the network. During prediction, new input is input into the trained model after dimension expansion according to the same rule, and a predicted value is obtained. According to the method, parameter arrangement symmetry is actively broken, and the loss function landscape is remarkably improved.

Description

Technical Field

[0001] This invention relates to the fields of artificial intelligence and high-efficiency computing technology, specifically to a one-dimensional function fitting method and system based on input dimension expansion. Background Technology

[0002] One-dimensional function fitting is a fundamental task in scientific computing and engineering practice, widely used in sensor calibration, real-time control signal generation, digital signal processing, and physical simulation. Traditional methods, such as polynomial interpolation and spline fitting, perform well when dealing with smooth, low-frequency functions, but often suffer from insufficient fitting accuracy, overfitting, or numerical instability when faced with one-dimensional functions containing high-frequency oscillations, discontinuities, or complex fractal structures.

[0003] In recent years, neural networks have attracted widespread attention for their powerful nonlinear fitting capabilities in approximating one-dimensional functions. While structures such as the Multilayer Perceptron (MLP) theoretically possess universal approximation properties, they still face the following prominent challenges in practical training and applications: First, the order of neurons within the same layer of a neural network does not affect the network output. This symmetry results in a large number of flat regions and equivalent minima in the loss function surface, making gradient-based first-order optimization algorithms prone to slow convergence and low training efficiency.

[0004] Secondly, during training, neural networks tend to fit the low-frequency components of the target function first, while the high-frequency components are often captured only in the later stages of training. For functions containing high-frequency harmonics, discontinuities, or non-smooth features, the network is prone to producing overly smoothed outputs, limiting the accuracy of the approximation.

[0005] Third, on resource-constrained platforms such as embedded systems, application-specific integrated circuits (ASICs), and field-programmable gate arrays (FPGAs), the available computing units, storage space, and power consumption budgets strictly limit the network size and inference complexity. Simply increasing the network width or introducing complex feature transformations may improve the fitting ability, but it often leads to a significant increase in the number of parameters and computational overhead, making it difficult to meet the dual requirements of real-time performance and low resource consumption.

[0006] To address the aforementioned issues, several improvements have emerged in existing technologies. For example, designing dedicated function fitting circuits and using piecewise polynomial approximation combined with lookup tables to achieve high-speed computation, but these methods suffer from poor versatility and high storage overhead. Hardware-friendly approximation computation or reusing computational resources can be employed to reduce implementation costs, but these methods are still limited by the ability to fit and express the data and the flexibility of the structure. Some studies have also attempted to alleviate the symmetry problem by introducing noise and using diversified initialization, but these methods are unstable and lack structural guidance.

[0007] In summary, existing methods often struggle to balance fitting accuracy, convergence speed, and inference resource consumption. In particular, they lack a systematic approach to proactively and structurally break parameter symmetry, improve loss patterns, and suppress spectral bias while strictly controlling the number of parameters and computational overhead. Therefore, a new method is urgently needed that is suitable for resource-constrained environments and can significantly improve the fitting performance of one-dimensional functions (especially high-frequency and non-smooth functions) without increasing the inference burden. Summary of the Invention

[0008] This invention provides a one-dimensional function fitting method based on input dimension expansion. Its purpose is to alleviate the spectral bias problem and flat convergence problem in high-frequency, non-smooth one-dimensional function fitting on a resource-constrained hardware platform without increasing inference overhead and total number of parameters, by actively breaking the symmetry of neural network parameter arrangement and reconstructing the loss landscape.

[0009] To achieve the above objectives, the first aspect of the present invention provides a one-dimensional function fitting method based on input dimension expansion, comprising the following steps: Obtain the one-dimensional objective function to be fitted and the set of training input scalars sampled within its domain; For each input scalar in the set of input scalars, an extended input vector is constructed according to a preset symmetry breaking rule, wherein the extended dimension is odd, the input scalar is located at the center of the vector, and the remaining components are filled with the same preset constant related to the domain of the objective function. Construct a neural network model with the extended input vector as input, and adjust the number of hidden layer neurons so that the deviation between the total number of trainable parameters of the expanded neural network model and the total number of trainable parameters of the baseline model with the original one-dimensional input does not exceed a set value. The extended input vector and its corresponding one-dimensional objective function value are input into the neural network model, and the parameters of the neural network model are trained using an optimization algorithm based on second-order derivative information. The one-dimensional input scalar to be predicted is mapped to an extended input vector according to the same symmetry-breaking input dimension expansion rule, and then input into the trained neural network model to obtain the predicted value of the one-dimensional objective function at that input.

[0010] Furthermore, in the step of constructing the extended input vector, the value of the preset constant is related to the definition interval of the one-dimensional objective function, and the value is the midpoint of the definition interval.

[0011] Furthermore, in the step of constructing the extended input vector, the extended dimension is 5.

[0012] Furthermore, when the definition interval of the one-dimensional objective function is When the preset constant takes the value .

[0013] Furthermore, the neural network model is a multilayer perceptron containing at least two hidden layers; after introducing extended input, the number of neurons in each hidden layer is reduced proportionally to the unextended baseline model, such that: the number of rows in the first layer weight matrix matches the extended dimension; and the total number of trainable parameters of the neural network model remains basically constant.

[0014] Furthermore, the optimization algorithm based on second-order derivative information is an L-BFGS-type quasi-Newton optimization algorithm.

[0015] Furthermore, the one-dimensional objective function is at least one of the following: a sinusoidal superposition function containing high-frequency harmonic components, a square wave or sawtooth wave function containing discontinuities, or a Weierstrass function that is continuous everywhere but not differentiable everywhere.

[0016] To achieve the above objectives, a second aspect of the present invention provides a one-dimensional function fitting system based on input dimension expansion, comprising: The data preprocessing module is used to obtain the one-dimensional objective function to be fitted and the set of training input scalars in its domain, and to construct each input scalar into an extended input vector according to a preset symmetry breaking rule, wherein the extended dimension is odd, the input scalar is located at the center of the vector, and the remaining components are filled with the same preset constant related to the domain of the objective function. The parameter-constrained neural network module is used to construct a neural network model with the extended input vector as input, and to adjust the number of hidden layer neurons so that the deviation between the total number of trainable parameters of the expanded model and the total number of trainable parameters of the baseline model with the original one-dimensional input does not exceed a set value. The training optimization module is used to input the extended input vector and its corresponding one-dimensional objective function value into the neural network model, and to train the model parameters using an optimization algorithm based on second-order derivative information. The inference module is deployed to map the one-dimensional input scalar to be predicted into an extended input vector according to the same symmetry-breaking input dimension expansion rule as in the training phase, and input it into the trained neural network model to obtain the predicted value of the objective function at that input.

[0017] To achieve the above objectives, a third aspect of the present invention provides an electronic device including a memory and a processor, the memory being used to store a program that supports the processor in executing the one-dimensional function fitting method based on input dimension expansion, and the processor being configured to execute the program stored in the memory.

[0018] To achieve the above objectives, a fourth aspect of the present invention provides a computer-readable storage medium storing a computer program, wherein the computer program, when executed by a processor, performs the steps of the one-dimensional function fitting method based on input dimension expansion.

[0019] The beneficial effects of this invention are: Compared with existing technologies, this invention provides a one-dimensional function fitting method based on input dimension expansion. It expands each one-dimensional input scalar according to an odd-dimensional centrosymmetry rule, mapping it to an expanded vector centered on the original input and filled with the same constants on both sides. This introduces a structured asymmetry into the input layer that cannot be eliminated by neuron substitution, directly breaking the parameter arrangement symmetry. Secondly, when constructing the neural network with the expanded vector as input, it strictly constrains the total number of trainable parameters to be basically consistent with the original baseline model. The redistribution of parameters from the deep layers to the input layer is achieved by adjusting the hidden layer width, ensuring that the multiplication and addition operations and storage overhead during the inference stage do not increase. Finally, an optimization algorithm based on second-order derivative information is used during training. The input structure constructed by the expanded vector, with its clear geometric meaning, guides the optimization path and reshapes the loss function landscape. This enables the network to more effectively capture high-frequency components and non-smooth features under conditions of limited parameter size and computational overhead, while accelerating convergence and improving fitting accuracy. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below.

[0021] Figure 1 This is a flowchart of a one-dimensional function fitting method based on input dimension expansion disclosed in an embodiment of the present invention.

[0022] Figure 2 This is a schematic diagram of input dimension expansion disclosed in an embodiment of the present invention.

[0023] Figure 3 This is a comparison of the convergence curves of F1 sine multi-frequency fitting and loss.

[0024] Figure 4 This is a comparison of the convergence curves of high-frequency edge fitting and loss for an F2 square wave.

[0025] Figure 5 This is a comparison of the convergence curves of high-frequency linear fitting and loss for an F3 sawtooth wave.

[0026] Figure 6 This is a comparison chart of convergence curves for F4 triangular wave fitting and loss.

[0027] Figure 7This is a comparison chart of the convergence curves of F5 modulated sine wave fitting and loss.

[0028] Figure 8 This is a comparison of the convergence curves of the F6 frequency chirp fitting and loss.

[0029] Figure 9 This is a comparison chart of convergence curves for F7 square wave duty cycle fitting and loss.

[0030] Figure 10 This is a comparison of the convergence curves of the F8 van der Bohr approximation and the loss.

[0031] Figure 11 This is a comparison of the convergence curves of fitting and loss for an F9 Weierstrass function that has nowhere to differentiate.

[0032] Figure 12 This is a comparison chart of convergence curves for F10 combined function fitting and loss. Detailed Implementation

[0033] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0034] According to embodiments of the present invention, it should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the following methods, in some cases the steps shown or described may be executed in a different order than that shown here.

[0035] like Figure 1 As shown, this invention provides a one-dimensional function fitting method based on input dimension expansion, comprising the following steps: Step S100: Obtain the one-dimensional objective function to be fitted and the set of training input scalars sampled within its domain; This step requires obtaining the one-dimensional objective function to be fitted (i.e., a function rule with a single real number as the independent variable and the output also being a real number) and the set of training input scalars sampled within the function's domain (i.e., the effective range of values ​​for the independent variable).

[0036] For example, in Example 1, the objective function is defined in the interval [0, 1]. Multi-frequency sine function Its mathematical expression is:

[0037] in, As the independent variable, The dependent variable; subsequently, within this domain The system obtains a series of input scalar values ​​through random or uniform sampling, forming a set of input scalars for training. ,in For the number of samples, each Both are real scalars. .

[0038] Step S200: For each input scalar in the set of input scalars, construct an extended input vector according to a preset symmetry breaking rule, wherein the extended dimension is odd, the input scalar is located at the center of the vector, and the remaining components are filled with the same preset constant related to the domain of the objective function. This step transforms the original one-dimensional scalar input into a high-dimensional vector through a structured mapping. This transformation is not simply about adding random features, but rather aims to proactively break the parameter arrangement symmetry that may exist within subsequent neural networks (especially the first layer), thereby improving the loss function landscape of the optimization process.

[0039] Specifically, for each input scalar in the training set This step constructs its corresponding extended input vector according to the following rules. : Determine the expansion dimension: Choose an odd number of at least 3 as the expansion dimension. . It can be represented as:

[0040] in, It is a positive integer, which means that the expanded vector will have a clear "center-sides" structure.

[0041] Constructing a vector structure: converting the original input scalar Placed in the extended vector The center position, that is, the ( ) components. Fill on both the left and right sides Each component has a set of values, and all of these filled components take the same preset constant. Therefore, the input vector is expanded. Represented as:

[0042] in, There are on both the left and right sides indivual ; To expand the dimension, it must be an odd number (such as 3, 5, 7, etc.), which determines the final length of the input vector. Depend on Defined half-side padding amount. The input value (a scalar) is the original, one-dimensional function to be fitted. This is a preset constant whose value is related to the domain of the objective function.

[0043] Preferably, The value is taken as the interval defined by the objective function. The midpoint, that is:

[0044] This choice makes the extended input numerically geometrically symmetric about the domain, enhancing the effect of symmetry breaking. For example, for a... The periodic function defined above, constant The preferred option is... .

[0045] It's important to note that "parameter arrangement symmetry" refers to the fact that in traditional multilayer perceptrons, if neurons within the same layer are swapped sequentially, the network's input-output mapping remains unchanged. This results in a large number of equivalent flat regions in the loss function. This step involves constructing... This fixed structure ensures that each input The position (center) in the vector and the constants of different positions. The combination relationship is unique and cannot be reproduced by neuron swapping, thus actively "breaking" this symmetry at the data input level and providing a clearer optimization path for the network.

[0046] In most application scenarios, in order to achieve a good balance between model complexity and symmetry breaking strength, the dimensionality is expanded. The preferred value is 5 (i.e.) = 2). At this point, the extended input vector is specifically constructed as follows:

[0047] For the definition in Periodic functions (such as sine combinations) can be directly taken as... .

[0048] The centrosymmetric vector structure formed by this dimension expansion operation can be found in [reference needed]. Figure 2 The diagram shows that Figure 2 Taking 2D extension as an example, the original scalar input is intuitively demonstrated. How to place it at the center, and with the constant components on both sides Together they form the extended input vector.

[0049] Step S300: Construct a neural network model with the extended input vector as input, and adjust the number of hidden layer neurons so that the deviation between the total number of trainable parameters of the expanded neural network model and the total number of trainable parameters of the baseline model with the original one-dimensional input does not exceed a set value. This step, under the premise of a basically fixed total computational resource (parameter count) budget, internally reallocates the network structure, thereby providing a matching computational graph for the symmetry-breaking structure introduced in step S200 without increasing inference overhead. The specific implementation process is as follows: First, we need to define a primitive one-dimensional scalar. Let be the baseline neural network model for input. This model has a specific number of layers and the number of neurons per layer, and its total number of trainable parameters (i.e., the sum of all weights and biases) is denoted as . The subsequent goal of this step is to construct a system with an expanded dimension as... vector For the new network as input, and to make its total number of trainable parameters Satisfy constraints:

[0050] in, Setting a small preset threshold (e.g., 5%) means that the size of the expanded model must be basically consistent with the baseline model, with the deviation controlled within ±5%.

[0051] To achieve the above constraints, since the input dimension increases from 1 dimension to... The weight matrix from the input layer to the first hidden layer naturally increases the number of parameters. To offset this increase, the width of subsequent hidden layers (i.e., the number of neurons per layer) needs to be reduced. For example, in Example 1, the base model's hidden layer configuration is [100, 100, 50, 50]. When constructing an extended-dimensional network with a 5-dimensional input, the number of neurons in these hidden layers is adjusted proportionally (e.g., adjusted to [95, 95, 48, 48], etc.) to ensure... .

[0052] The constructed neural network model is a multilayer perceptron, containing one input layer (dimension 1). It consists of at least two hidden layers and one output layer (with a dimension of 1). The weight matrix of the first hidden layer has a dimension of [number of hidden layer neurons, ...]. This allows the model to match the expanded input dimension. Through this width adjustment, the total number of trainable parameters remains essentially constant, ensuring that the number of multiply-accumulate operations and model storage requirements during the inference (prediction) phase are on the same order of magnitude as the baseline model, thus avoiding hardware resource constraints.

[0053] This "structural redistribution under constant parameter constraints" is the core mechanism by which this invention improves performance without increasing inference costs. It "transfers" a portion of the parameters from the deeper layers of the network to the input layer and its immediate neighbors, allowing the weights at the network's front end to be consistent with the constants in the input vector. They interact to form a unique computational pattern equivalent to a refined combination of biases, which cannot be obtained through conventional bias terms or random feature extensions.

[0054] Step S400: Input the extended input vector and its corresponding one-dimensional objective function value into the neural network model, and train the neural network model parameters using an optimization algorithm based on second-order derivative information; Under the constraint of a limited total number of parameters, this step utilizes a matching optimization strategy to fully leverage the loss landscape improvement brought about by input dimensionality expansion, enabling the network parameters to converge quickly and accurately to approximate the objective function.

[0055] The extended input vector generated in step S200 The features used as training samples are their corresponding one-dimensional function values ​​calculated from the original objective function. These serve as labels for the samples. This constitutes the training sample pairs. The entire training set consists of multiple such sample pairs, which drive the model to learn the mapping from the extended input to the target output.

[0056] This step employs an optimization algorithm based on second-order derivative information. Specifically, the L-BFGS-like quasi-Newton optimization algorithm is preferred. Compared to common first-order optimizers (such as SGD and Adam), L-BFGS exhibits faster convergence speed and higher accuracy when solving small-to-medium-scale deterministic optimization problems with good smoothness. In this embodiment, the learning rate is set to 1.0, and the maximum number of iterations is set to 500.

[0057] The training process aims to minimize a preset loss function. This invention preferably uses mean squared error (MSE) as the loss function. Minimizing MSE means driving the model's predictions to be as close as possible to the true values ​​overall.

[0058] Driven by the optimization algorithm, the model parameters are iteratively updated. During training, key metrics need to be monitored to evaluate the convergence status, such as recording the decline curve of the loss function value with the number of iterations. Simultaneously, convergence criteria can be set, such as when the relative change in the norm of the gradient (or the rate of change of the loss value) is less than a very small threshold (e.g., 10). -10 When the MSE on the test set first falls below a preset precision threshold, it is considered converged. In some embodiments, the number of iterations required to first reduce the MSE on the test set below a preset precision threshold is also recorded to quantify the improvement in convergence speed.

[0059] The second-order optimization algorithm selected in this step synergizes with the "symmetry breaking" input structure in step S200. The expanded-dimensional input reconstructs the loss landscape in the parameter space, reducing flat areas; while algorithms such as L-BFGS can more effectively utilize this improved landscape geometry to find a better descent path. This facilitates the capture of difficult-to-fit features such as high-frequency components and discontinuities, effectively alleviating the spectral bias problem and achieving higher fitting accuracy within a limited number of iterations.

[0060] The convergence acceleration effect of this method on typical functions was intuitively verified through comparative experiments with different expansion dimensions; such as Figures 3-12 As shown, Figures 3-12 The convergence curves of different extended dimensions (1D, 3D, 5D, 7D) on typical functions are compared. Compared with the original 1D input, the odd-dimensional extended input structure proposed in this invention results in a faster decline in the convergence curve of the loss function and a lower error plateau. This directly confirms the effectiveness of input dimension expansion in improving the loss landscape and accelerating training convergence under constant parameter constraints.

[0061] Step S500: Map the one-dimensional input scalar to be predicted to an extended input vector according to the same symmetry-breaking input dimension expansion rule, and input it into the trained neural network model to obtain the predicted value of the one-dimensional objective function at that input.

[0062] The trained model is applied to a real one-dimensional function prediction task to give an estimate of the function value for any new input scalar. Since the parameter constant constraint is applied in step S300, the total number of parameters and network depth of the trained model are basically the same as those of the benchmark model. Therefore, the number of multiply-accumulate (MAC) operations consumed in a single inference is on the same order of magnitude as that of the benchmark model, without introducing additional computational burden.

[0063] To facilitate engineering applications, this invention further proposes a corresponding system implementation scheme, including: The data preprocessing module is used to obtain the one-dimensional objective function to be fitted and the set of training input scalars in its domain, and to construct each input scalar into an extended input vector according to a preset symmetry breaking rule, wherein the extended dimension is odd, the input scalar is located at the center of the vector, and the remaining components are filled with the same preset constant related to the domain of the objective function. The parameter-constrained neural network module is used to construct a neural network model with the extended input vector as input, and to adjust the number of hidden layer neurons so that the deviation between the total number of trainable parameters of the expanded model and the total number of trainable parameters of the baseline model with the original one-dimensional input does not exceed a set value. The training optimization module is used to input the extended input vector and its corresponding one-dimensional objective function value into the neural network model, and to train the model parameters using an optimization algorithm based on second-order derivative information. The inference module is deployed to map the one-dimensional input scalar to be predicted into an extended input vector according to the same symmetry-breaking input dimension expansion rule as in the training phase, and input it into the trained neural network model to obtain the predicted value of the objective function at that input.

[0064] According to another aspect of the embodiments of this application, an electronic device is also provided, including a processor and a memory, wherein the processor is configured to implement the steps of the method when executing a computer program stored in the memory.

[0065] In the above embodiments of the present invention, the descriptions of each embodiment have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions of other embodiments.

[0066] In the several embodiments provided in this application, it should be understood that the disclosed technical content can be implemented in other ways. The device embodiments described above are merely illustrative; for example, the division of units can be a logical functional division, and in actual implementation, there may be other division methods. For instance, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the displayed or discussed mutual coupling, direct coupling, or communication connection may be through some interfaces; the indirect coupling or communication connection between units or modules may be electrical or other forms.

[0067] Furthermore, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0068] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, read-only memory (ROM), random access memory (RAM), portable hard drives, magnetic disks, or optical disks.

[0069] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A one-dimensional function fitting method based on input dimension expansion, characterized in that, Includes the following steps: Obtain the one-dimensional objective function to be fitted and the set of training input scalars sampled within its domain; For each input scalar in the set of input scalars, an extended input vector is constructed according to a preset symmetry breaking rule, wherein the extended dimension is odd, the input scalar is located at the center of the vector, and the remaining components are filled with the same preset constant related to the domain of the objective function. Construct a neural network model with the extended input vector as input, and adjust the number of hidden layer neurons so that the deviation between the total number of trainable parameters of the expanded neural network model and the total number of trainable parameters of the baseline model with the original one-dimensional input does not exceed a set value. The extended input vector and its corresponding one-dimensional objective function value are input into the neural network model, and the parameters of the neural network model are trained using an optimization algorithm based on second-order derivative information. The one-dimensional input scalar to be predicted is mapped to an extended input vector according to the same symmetry-breaking input dimension expansion rule, and then input into the trained neural network model to obtain the predicted value of the one-dimensional objective function at that input.

2. The one-dimensional function fitting method based on input dimension expansion as described in claim 1, characterized in that, In the step of constructing the extended input vector, the value of the preset constant is related to the definition interval of the one-dimensional objective function, and the value is the midpoint of the definition interval.

3. The one-dimensional function fitting method based on input dimension expansion as described in claim 1 or 2, characterized in that, In the step of constructing the extended input vector, the extended dimension is 5.

4. The one-dimensional function fitting method based on input dimension expansion as described in claim 1, characterized in that, When the definition interval of the one-dimensional objective function is When the preset constant takes the value .

5. The one-dimensional function fitting method based on input dimension expansion as described in claim 1, characterized in that, The neural network model is a multilayer perceptron containing at least two hidden layers. After the extended input is introduced, the number of neurons in each hidden layer is reduced proportionally to the unextended baseline model, such that: the number of rows in the first layer weight matrix matches the extended dimension; and the total number of trainable parameters of the neural network model remains basically constant.

6. The one-dimensional function fitting method based on input dimension expansion as described in claim 1, characterized in that, The optimization algorithm based on second-order derivative information is an L-BFGS-type quasi-Newton optimization algorithm.

7. The one-dimensional function fitting method based on input dimension expansion as described in claim 1, characterized in that, The one-dimensional objective function is at least one of the following: a sinusoidal superposition function containing high-frequency harmonic components, a square wave or sawtooth wave function containing discontinuities, or a Weierstrass function that is continuous everywhere but not differentiable everywhere.

8. A one-dimensional function fitting system based on input dimension expansion, characterized in that, include: The data preprocessing module is used to obtain the one-dimensional objective function to be fitted and the set of training input scalars in its domain, and to construct each input scalar into an extended input vector according to a preset symmetry breaking rule, wherein the extended dimension is odd, the input scalar is located at the center of the vector, and the remaining components are filled with the same preset constant related to the domain of the objective function. The parameter-constrained neural network module is used to construct a neural network model with the extended input vector as input, and to adjust the number of hidden layer neurons so that the deviation between the total number of trainable parameters of the expanded model and the total number of trainable parameters of the baseline model with the original one-dimensional input does not exceed a set value. The training optimization module is used to input the extended input vector and its corresponding one-dimensional objective function value into the neural network model, and to train the model parameters using an optimization algorithm based on second-order derivative information. The inference module is deployed to map the one-dimensional input scalar to be predicted into an extended input vector according to the same symmetry-breaking input dimension expansion rule as in the training phase, and input it into the trained neural network model to obtain the predicted value of the objective function at that input.

9. An electronic device, comprising a memory and a processor, characterized in that, The memory is used to store a program that supports the processor in executing any of the one-dimensional function fitting methods based on input dimension expansion as described in claims 1-7, wherein the processor is configured to execute the program stored in the memory.

10. A computer-readable storage medium storing a computer program thereon, characterized in that, When the computer program is run by the processor, it executes the steps of any one of the one-dimensional function fitting methods based on input dimension expansion as described in claims 1-7.

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