PINN-based infrared window surface anti-reflection micro-nano structure optimization design method

Abstract

The invention discloses a PINN-based infrared window surface anti-reflection micro-nano structure optimization design method, which is oriented to a sub-wavelength microstructure (such as a square column array or a cylinder array) on the surface of an infrared optical window, and comprises the following steps of: integrating physical constraints into a neural network model; and high-precision prediction of the reflectivity of the micro-nano structure and intelligent optimization of structural parameters are realized. Aiming at the characteristics of data scarcity and high dimension of design parameters, the invention provides innovative designs such as a parameter space sampling strategy, a PINN model architecture, a physical constraint fusion strategy and a training optimization method, and can effectively solve the problems of long time consumption, large data demand, poor model generalization performance and the like of a traditional design method. And infrared anti-reflection micro-nano structure parameters which meet physical rules and are excellent in performance can be obtained under the condition of less simulation data. According to the technical scheme, intelligentization and high efficiency of infrared window anti-reflection micro-nano structure parameter design are achieved.

Description

Technical Field

[0001] This invention belongs to the field of micro-nano structure design and relates to a design method for infrared functional surface micro-nano structures, specifically a method for optimizing the design of infrared window surface anti-reflection micro-nano structures based on physical information neural network (PINN). Background Technology

[0002] Infrared windows are crucial optical components in infrared thermal imaging systems, and their surfaces often require anti-reflection treatment to minimize infrared light reflection loss at the interface and improve transmittance. Common anti-reflection methods include depositing multilayer anti-reflection films and fabricating subwavelength micro / nano structures on the surface. Compared to traditional thin films, micro / nano structure anti-reflection layers offer advantages such as broadband anti-reflection and good environmental stability, thus attracting widespread attention. However, the design and optimization of micro / nano anti-reflection structures heavily rely on electromagnetic simulations, typically employing the finite-difference time-domain (FDTD) method or rigorous coupled-wave analysis (RCWA) to calculate the full-band optical response for given structural parameters. This FDTD-based design method has significant bottlenecks: each simulation is time-consuming, and numerous simulations are required to find the optimal structural combination in high-dimensional parameter spaces, resulting in a lengthy design cycle. Furthermore, obtaining high-precision simulation data is costly, and batch fabrication and measurement of different structures are also very time-consuming in experiments.

[0003] Existing technologies have explored using machine learning or deep learning models to replace simulation for accelerated design. However, purely data-driven models have inherent limitations: due to the lack of physical constraints, they often only make reliable predictions within the range of the training data distribution. When the data exceeds the available range, they may give results that do not conform to physical laws, such as predicting reflectivity exceeding 100%, which is an incomprehensible phenomenon. Therefore, how to construct anti-reflection structure design models that combine physical consistency and generalization ability under conditions of scarce data has become a pressing technical problem to be solved in this field. Summary of the Invention

[0004] This invention provides an optimized design method for infrared window surface antireflection micro / nanostructures based on PINN. This method targets subwavelength microstructures (such as square or cylindrical arrays) on infrared optical window surfaces. By integrating physical constraints into a neural network model, it achieves high-precision prediction of the reflectivity of micro / nanostructures and intelligent optimization of structural parameters. Addressing the challenges of data scarcity and high-dimensional design parameters, this invention proposes innovative designs including a parameter space sampling strategy, a PINN model architecture, a physical constraint fusion strategy, and training optimization methods. These effectively solve the problems of long processing times, large data requirements, and poor model generalization performance associated with traditional design methods. It achieves infrared antireflection micro / nanostructure parameters that satisfy physical laws and exhibit excellent performance even with limited simulation data.

[0005] The objective of this invention is achieved through the following technical solution: An optimized design method for anti-reflection micro / nano structures on infrared window surfaces based on PINN includes the following steps: Step (1) Formulate a parameter space sampling strategy based on physical priors: By using theoretical analysis and engineering constraints, reasonable ranges for structural geometry type S, period T, height H, duty cycle f, operating wavelength λ, and incident angle θ are determined. Differentiated sampling methods are then employed for each parameter to cover key regions of the high-dimensional parameter space with a limited number of samples. The specific steps are as follows: (a) Period T: Based on the equivalent medium theory, the upper limit constraint of the period is determined. Considering the processing limitations, the actual period range is extended to [0.5~5μm]. Within this range, importance sampling is adopted, and an exponential decay weight function is introduced to make the samples distributed in the sensitive interval with high information density, ensuring sufficient sampling of key small period regions, while avoiding unnecessary sampling of large period regions. (b) Height H and duty cycle f: First, the theoretical optimal range of height H is determined using the film resonance condition. Combined with the processing depth capability, the height is limited to 0.1~3μm. Within this range, height sample points are generated by Latin hypercube sampling. Then, for each height sample, the corresponding optimal duty cycle f is calculated according to the resonance condition formula, and additional dense sampling is performed in its neighborhood to obtain a refined value of the duty cycle, thus forming a non-uniform two-dimensional sampling grid around the theoretical resonance condition. (c) Wavelength λ: For the transmission band of the material, discrete wavelength points are obtained by uniform sampling. These wavelength sampling points uniformly cover the working band of the infrared window to ensure that the training data covers the typical response of the entire spectral range. (d) Angle of incidence θ: Angle samples are selected by uniformly distributing the cosine value of the angle of incidence within the range of [0°~60°]; Step (2) Constructing the PINN model architecture that incorporates physical priors The PINN model architecture is based on the PINN framework and consists of an input layer, a hidden layer, and an output layer, wherein: The input layer takes five continuous parameters—period T, height H, duty cycle f, wavelength λ, and incident angle θ—and one discrete structure type parameter G as input. To accommodate multimodal input, the continuous parameters are first normalized to a uniform range of [0,1]. For the structure type parameter G, one-hot encoding is used to encode the columnar array into a two-dimensional vector of (1,0) or (0,1). Finally, the input layer maps the processed 6-dimensional feature vector to the hidden layer through a fully connected layer, serving as the starting point for subsequent network computations. The hidden layer is a 5-layer fully connected structure, with 128 neurons in each layer, and the Swish activation function is introduced. A wavelength-period attention mechanism is introduced in the 3rd hidden layer. The output layer adopts a dual-channel structure to predict the reflectivity of S-polarization and P-polarization respectively. An energy conservation hard constraint module is embedded at the output end, and a constructive projection method is used to force the output to strictly satisfy the energy conservation. Step (3) Embed physical constraints into the PINN model: To further ensure the physical reliability of the model's predictions, multiple electromagnetic physical constraints, such as the law of conservation of energy, reciprocity symmetry, and wavelength-period resonance conditions, are explicitly integrated into the model training process. The specific steps are as follows: (a) Hard constraint of energy conservation: Energy conservation is directly achieved through the network output layer; (b) Reciprocity symmetric constraint: On the one hand, during dataset preprocessing, for each original sample Add an extra "mirror" sample The data is input into the network in pairs for training; on the other hand, a symmetry penalty term is introduced into the loss function of the model training, that is, the square of the difference between the two predicted reflectances obtained from the mirror input is calculated and added to the total loss with weight; (c) Resonance gradient soft constraint: For parameter region samples that theoretically have resonance, the gradient of their reflectance with respect to wavelength is pre-calculated. During training, in addition to minimizing the error between the predicted value and the true value, the difference between the gradient of the model prediction curve with respect to wavelength and the theoretical gradient is further minimized. To ensure the effectiveness of gradient penalty, automatic differentiation is used to obtain the derivative of the PINN output with respect to wavelength, and the gradient is subjected to Gaussian filtering for noise reduction and smoothing. Step (4) Phased and gradual training: The training process is divided into three stages: pre-training, physical fine-tuning, and fine-tuning. Constraints are gradually introduced to balance model accuracy and physical consistency. The specific steps are as follows: (a) Pre-training phase: The network is pre-trained using a purely data-driven approach; (b) Physical fine-tuning stage: After obtaining the pre-trained model, physical constraints are gradually added to fine-tune the model; (c) Fine-tuning stage: The L-BFGS second-order optimizer is used to fine-tune the model to obtain the final PINN model; Step (5) Structural optimization: The structure optimization design is performed using the trained PINN model, and the specific steps are as follows: (a) Discretize the continuous parameters that need to be optimized within the reasonable range of values ​​determined in step (1) according to a preset fixed step size, thereby transforming the infinite continuous parameter space into a finite, multidimensional discrete parameter grid. (b) Traverse each node on the grid. Each node represents a specific combination of structural parameters. For each combination in the grid, use the trained PINN model as an efficient surrogate model to quickly calculate its average reflectivity within the preset target working band and incident angle range, and use this as the performance evaluation value of the parameter combination. (c) After completing the traversal and evaluation of all grid points, the node with the lowest performance evaluation value is identified by comparison, and its corresponding parameter combination is determined. The final optimized design was selected and will be used directly to guide subsequent micro- and nano-fabrication and implementation. Step (6) Structure Implementation: Based on the optimized parameters, anti-reflection micro / nano structures were fabricated on the surface of the infrared window using micro / nano fabrication techniques. The specific steps are as follows: (a) The infrared window substrate is directly clamped onto the high-precision air-bearing spindle of an ultra-precision machine tool; (b) Based on the optimal parameters determined in step (5), a tool path is generated, and a diamond tool is driven by a Fast Tool Servo (FTS) system to precisely remove material from the high-speed rotating window surface; (c) After turning, the tiny debris and coolant generated during the machining process are thoroughly removed through a cleaning process to obtain a finished infrared window with anti-reflective micro-nano structures directly integrated on the surface.

[0006] This invention, through the above technical solution, achieves intelligent and efficient design of infrared window anti-reflection micro / nano structure parameters, and achieves significant results compared with traditional methods: (a) Significantly improved simulation accuracy and efficiency: The trained PINN model can predict the reflectance spectrum of a given combination of structural parameters within milliseconds, and its prediction results are in high agreement with the high-precision RCWA simulation values. The average reflectance prediction error on the test set is only a few percentage points, and the maximum error is reduced by about 31.2% compared with the model without physical constraints. This is equivalent to achieving near-full-wave electromagnetic simulation accuracy at extremely low computational cost, significantly shortening the design iteration cycle.

[0007] (b) Significantly Enhanced Physical Consistency: Due to the integration of constraints such as energy conservation, reciprocity, and resonant gradients, the physical consistency index of the model in this invention is far superior to that of traditional data-driven models. For example, after the hard constraint of energy conservation, the reflectivity predicted by the model is strictly... The energy conservation violation rate was reduced to 0; the reciprocity constraint ensured that the predictions at all angles met symmetry, with the bias approaching zero; and the resonance gradient constraint ensured the accurate location of the abrupt change in the reflectivity spectrum, reproducing the physical resonance phenomenon. Overall, the physical constraint scheme reduced the model's total physical violation rate by more than 20%. Therefore, the model's predictions of unknown parameters are more reliable and trustworthy, avoiding the non-physical results that might have occurred with previous pure data models.

[0008] (c) Reduced data requirements and improved generalization ability: This invention, through physical prior-guided sampling and data augmentation, trains a high-performance model with only about 30,000 samples, reducing the required data volume by several times compared to training ordinary deep networks. Simultaneously, due to the inclusion of physical constraints, the model can correctly extrapolate physical laws, maintaining good accuracy even in parameter regions not covered by the training samples. Especially under out-of-sample conditions such as large incident angles and extreme parameter combinations, it can provide reasonable predictions, eliminating the dependence of traditional models on the distribution of training data. This means that the method of this invention has good generalization and applicability, adapting to the anti-reflection design requirements of different infrared bands and structural types.

[0009] (d) Design Performance Optimization: Utilizing the fast and accurate PINN model, structural parameter combinations superior to traditional empirical designs can be efficiently searched. For example, the square columnar microstructure array designed on a ZnS infrared window in this invention reduces the average reflectivity to below 5% (single-sided) in the 4–13 μm band, a significant improvement compared to an unstructured substrate (reflectivity approximately 15%). Therefore, the method of this invention not only accelerates the design process but also finds a superior anti-reflection solution, significantly improving the transmittance performance of the infrared window.

[0010] In summary, this invention integrates physical principles into deep learning, achieving high-precision, physically consistent, and highly generalizable intelligent design of anti-reflection micro / nano structures. This method reduces repeated manual trial and error and extensive simulation calculations, demonstrating significant practical value and promising prospects for the design of microstructures on the surface of infrared optical components. Attached Figure Description

[0011] Figure 1 This is a schematic diagram of the PINN network structure; Figure 2 Flowchart of input-output mapping for the PINN model; Figure 3 A schematic diagram of a high-dimensional parameter space sampling strategy; Figure 4 This is a schematic diagram of the physical constraint fusion module; Figure 5 This is a schematic diagram of a hybrid training strategy; Figure 6This is a flowchart of an optimized design method for anti-reflection micro / nano structures on infrared window surfaces based on PINN. Detailed Implementation

[0012] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0013] This invention provides an optimized design method for anti-reflection micro / nano structures on infrared window surfaces based on PINN, such as... Figure 6 As shown, the method includes the following steps: Step (1) Formulate a parameter space sampling strategy based on physical priors: To construct a high-quality training dataset covering key physical phenomena, this invention first formulates a hybrid parameter sampling strategy based on physical priors. Infrared anti-reflection microstructure design involves six parameters (structural geometry type S, period T, height H, duty cycle f, operating wavelength λ, and incident angle θ), and direct uniform sampling is inefficient. This invention utilizes theoretical analysis and engineering constraints to determine reasonable ranges for each parameter and employs differentiated sampling methods for each parameter, covering key regions of the high-dimensional parameter space with a limited number of samples. (a) Period T: The upper limit constraint of the period is determined based on the equivalent medium theory. When the incident wavelength is at its minimum, the theoretical maximum period is approximately a certain value. Considering machining constraints (such as the radius of single-point turning tools and machine tool accuracy), the actual period range is extended to [0.5~5μm]. Within this range, importance sampling is adopted, and an exponentially decaying weight function is introduced so that about 70% of the samples are distributed in the sensitive interval with high information density, ensuring sufficient sampling of key small period regions while avoiding unnecessary sampling of large period regions.

[0014] (b) Height H and Duty Cycle f: Height and duty cycle are strongly coupled. This invention employs a layered progressive sampling method: First, the theoretical optimal range of height H is determined using the film resonance condition (e.g., a range satisfying an optical path difference of approximately half a wavelength), and the height is limited to 0.1~3μm based on processing depth capabilities. Within this range, 30 height sample points are generated using Latin hypercube sampling. For each height sample, the corresponding optimal duty cycle f is calculated according to the resonance condition formula, and additional dense sampling is performed in its neighborhood to obtain a refined value of the duty cycle (approximately several f values ​​are taken near each height). This forms a non-uniform two-dimensional sampling grid around the theoretical resonance condition, obtaining approximately 10 representative duty cycle samples. This strategy fully considers the physical correlation between H and f, concentrating the samples in the highly sensitive resonance region.

[0015] (c) Wavelength λ: For the transmission band of ZnS material (approximately 4~13 μm), 15 discrete wavelength points were obtained by uniform sampling. These wavelength sampling points uniformly cover the working band of the infrared window, ensuring that the training data covers the typical response of the entire spectral range.

[0016] (d) Incident angle θ: Considering the angle dependence of the optical response under oblique incidence conditions, this invention uses cosine-weighted sampling for the incident angle. Six angle samples are selected by uniformly distributing the cosine value of the incident angle within the range of [0°~60°], and the sampling density is appropriately increased in the large angle region. This cosine-distributed sampling method makes the obtained samples closer to the incident distribution characteristics of light in the actual optical system.

[0017] Using the above strategy, approximately 30,000 sets of sample data with different parameter combinations were collected for model training. The sampling results were verified to have good uniform coverage and representativeness: the parameters are basically independent (verified by mutual information analysis), and the samples in any two dimensions are evenly distributed on the projection plane without missing any important regions. This hybrid sampling scheme that incorporates physical constraints significantly improves the efficiency of parameter space exploration while ensuring engineering feasibility, providing a high-quality data foundation for subsequent model training. Figure 3 This is a schematic diagram of the high-dimensional parameter space sampling strategy used in this invention. The diagram lists the value ranges and sampling methods of each design parameter, including importance sampling of period T, Latin hypercube sampling of height H, resonant neighborhood sampling of duty cycle f, uniform sampling of wavelength λ, and cosine-weighted sampling of incident angle θ. The above sampling methods together generate a highly representative dataset for model training.

[0018] Step (2) Construct the PINN model architecture that incorporates physical priors: This invention constructs a neural network architecture that integrates physical priors to establish a mapping relationship between micro / nano structure parameters and reflectivity. The network is designed using a Physical Information Neural Network (PINN) framework, consisting of an input layer, hidden layers, and an output layer, and is specifically improved to address the characteristics of anti-reflective microstructures. (a) Input Layer Design: The input includes five continuous parameters (period T, height H, duty cycle f, wavelength λ, and incident angle θ) and one discrete structure type parameter (e.g., square prism or cylinder). To accommodate multimodal inputs, the continuous parameters are first normalized to a uniform range of [0,1]. This eliminates numerical instability caused by different dimensions and makes the gradient update scales of each parameter similar. For the structure type parameter G, this invention uses one-hot encoding to encode the square prism and cylinder types as two-dimensional vectors of (1,0) or (0,1), respectively. This encoding method preserves information about different geometric topologies, enabling the network to differentiate between different structural shapes. Finally, the input layer maps the processed 6-dimensional feature vector to the hidden layer through a fully connected layer, serving as the starting point for subsequent network computations.

[0019] (b) Hidden Layer Design: The network hidden layers are multi-layer fully connected structures used to extract the nonlinear relationship between parameters and reflectivity. This invention uses 5 hidden layers, each containing 128 neurons, and introduces the Swish activation function (…). (β is trainable). The continuity of the second derivative of the Swish function helps improve the smoothness and gradient stability of reflectance prediction. To further integrate the characteristics of optical physics, a wavelength-period attention mechanism is introduced in the third hidden layer. This attention module assigns different weights to the hidden feature channels according to the combination of wavelength λ and period T, as shown in the formula: (where α is a trainable weight), enabling the network to automatically focus on relevant features when optical resonance conditions occur (specific λ and T cause abrupt changes in reflection). This mechanism helps the network capture thin-film interference resonance phenomena, corresponding to the wavelength-period coupling effect in actual physics.

[0020] (c) Output Layer Design: Considering that reflectance prediction must meet optical physical constraints, the output layer adopts a dual-channel structure to predict the reflectance of S-polarization and P-polarization respectively. Specifically, the features of the last hidden layer are first mapped to the scalar output through independent fully connected layers, and then a Sigmoid activation is applied to constrain it to the range of [0,1], ensuring that the predicted value is within a reasonable physical range. Furthermore, this invention embeds an energy conservation hard constraint module at the output end, using a constructive projection method to force the output to strictly satisfy energy conservation (reflectance not greater than 1). This is achieved by applying a Sigmoid output... Applying the projection operator: ,in This is a numerically stable term. The final output obtained after this transformation... Strictly meet Furthermore, the sum of the two polarization channels does not exceed 1, thus ensuring energy conservation. Unlike penalties in the loss function, applying hard constraints directly to the structure completely eliminates physical violations. After being processed by this constraint, the output layer's dual channels output predicted values ​​for S-polarization and P-polarization reflectivity.

[0021] In summary, the constructed PINN model architecture integrates innovative designs such as geometric prior encoding (one-hot input), a dedicated attention mechanism, and a hard physics-constrained output layer. Through these improvements, the network possesses the ability to automatically learn electromagnetic wave resonance characteristics and strictly adhere to physical laws. According to feature visualization analysis, the network's hidden layers have autonomously learned representations consistent with the laws of electromagnetic propagation: for example, in the third layer containing attention, the activation spectrum exhibits a striped distribution with periodic variations depending on wavelength, corresponding to the Fabry-Pérot type resonance phenomenon. This indicates that the model of this invention has stronger physical meaning compared to traditional black-box neural networks, and both interpretability and prediction accuracy are improved.

[0022] Figure 1 This is a schematic diagram of the PINN network structure, showing the input parameters, hidden layers (including an attention mechanism), and a dual-channel structure with energy conservation constraints on the output layer. This invention employs a neural network with 5 fully connected hidden layers, each containing 128 neurons. A wavelength-period attention mechanism is introduced in the third layer to highlight resonance characteristics. The output layer uses dual channels to predict the reflectivity of S-polarization and P-polarization respectively, and embeds an energy conservation hard constraint module to ensure that the output conforms to physical laws. Figure 2 The flowchart shows the input-output mapping process of the PINN model. The model takes micro / nano structure parameters (period T, height H, duty cycle f, structure type, etc.) and operating conditions (wavelength λ, incident angle θ) as inputs and outputs the predicted S-polarization and P-polarization reflectance under the corresponding conditions.

[0023] Step (3) Physical constraint fusion: To further ensure the physical reliability of the model's predictions, this invention explicitly integrates various electromagnetic physical constraints into the model training process, including: the law of conservation of energy, reciprocity symmetry, and wavelength-period resonance conditions. These physical priors are embedded through a combination of hard and soft constraints, enabling the resulting PINN model to not only fit the data but also "understand" the underlying physical laws. The specific strategy is as follows: (a) Hard constraint on energy conservation: As mentioned above, energy conservation is directly achieved through a special design of the network output layer (output projection transformation). Compared to soft constraint methods that only add a penalty term to the loss function, hard constraints ensure that the model always meets the requirement that reflectivity does not exceed 100%. During training, it was observed that the model with this hard constraint never violated energy conservation, while the unconstrained model occasionally predicted non-physical values ​​where the sum of reflectivity was slightly greater than 1.

[0024] (b) Reciprocity Symmetry Constraint: According to the optical reciprocity theorem, when the structure is symmetrical in all directions, exchanging the incident and outgoing directions should not affect the reflectivity. Therefore, this invention proposes a dual strategy at the data level and the loss level to achieve the reciprocity constraint: on the one hand, during dataset preprocessing, for each original sample... Add an extra "mirror" sample (Or equivalently, samples incident from the other side) form paired data inputs for network training. On the other hand, a symmetry penalty term is introduced into the model training loss function; that is, the square of the difference between the two predicted reflectances obtained from mirrored inputs is calculated and weighted and added to the total loss. Through these two measures, the model is forced to learn to output the same reflectance value under both incident and reverse incident conditions. Experiments show that after symmetry-constrained training, the model's reflectance at each incident angle satisfies... The originally unconstrained model in Even with significant asymmetry, the error was reduced to almost zero. Furthermore, the introduction of mirrored samples effectively doubled the amount of training data, further enhancing the model's generalization ability across different perspectives.

[0025] (c) Resonance Gradient Soft Constraint: For the resonance phenomenon where the reflectivity of micro / nanostructures abruptly changes near a specific wavelength, this invention introduces gradient matching loss to help the model learn this physical law. Specifically, for samples in parameter regions where resonance theoretically exists, the gradient of reflectivity with respect to wavelength is pre-calculated (approximate gradient values ​​can be obtained through two-dimensional thin-film interference formulas or high-precision numerical differencing). During training, in addition to minimizing the error between the predicted and true values, the difference between the model's predicted curve gradient with respect to wavelength and the theoretical gradient is further minimized. This penalty prompts the network to correctly fit the steep drop in reflectivity in the resonance region, rather than simply fitting values ​​at discrete points. To ensure the effectiveness of the gradient penalty, this invention uses automatic differentiation to obtain the derivative of the PINN output with respect to wavelength and performs Gaussian filtering for noise reduction and smoothing of the gradient. After gradient constraint training, the model's prediction accuracy near the resonance wavelength is significantly improved, the position and width of the reflectivity peaks / valleys match the physical expectations, and the prediction error is reduced by approximately 30% compared to when this constraint is not applied.

[0026] By combining the above constraint fusion strategies, the PINN model of this invention simultaneously receives feedback from both data errors and physical violations during training, gradually approximating a solution that both fits the data and satisfies physical laws. The synergistic effect of multiple constraints improves the overall performance of the model. According to constraint effect analysis, adding gradient penalty can further increase the satisfaction rate of energy conservation constraints by about 2 percentage points, demonstrating a positive coupling effect between different physical priors. This invention, through this explicit embedding of physical knowledge, makes the model's prediction of electromagnetic behavior more reliable and self-consistent.

[0027] Figure 4 This diagram illustrates the physical constraint fusion module, which combines physical priors with a data-driven model to form the PINN anti-reflection design model of this invention. The diagram demonstrates the integration of physical laws into the data-driven model: based on a purely data-driven neural network, hard constraints of energy conservation, as well as soft constraints of reciprocity and resonance, are introduced and fused to form a Physical Information Neural Network (PINN) model, thereby ensuring that the prediction results both fit the data and conform to physical laws.

[0028] Step (4) Training optimization: Since the model of this invention simultaneously incorporates data fitting objectives and multiple physical constraints, direct end-to-end training may lead to unstable optimization or a trade-off dilemma. Therefore, this invention designs a phased hybrid training strategy, dividing the training process into three stages: pre-training, physical fine-tuning, and precision tuning, gradually introducing constraints and balancing model accuracy with physical consistency. (a) Pre-training stage: The network is first pre-trained using a purely data-driven approach. In this stage, only the ground truth reflectance values ​​obtained from simulation are used for supervision (the loss function is the standard mean squared error, MSE), without introducing physical constraints. The Adam optimizer is used for approximately 100 epochs to allow the model to quickly learn the basic input-output mapping relationship. To improve convergence speed, simple samples (such as perpendicularly incident, small-period structures) are used preferentially in the early stages of pre-training. After the model converges, complex samples (samples with large angles and resonance regions) are gradually added. This allows the network parameters to find better initial solution positions, preparing for the subsequent introduction of physical constraints.

[0029] (b) Physical Fine-tuning Stage: After obtaining the pre-trained model, the aforementioned physical constraints are gradually added to fine-tune the model. The loss function in this stage is the weighted sum of data error and each physical penalty term. To avoid manually adjusting the weight ratio, this invention adopts an adaptive weight balancing strategy: dynamically adjusting the coefficients of each loss term to make their gradient magnitudes comparable, thereby smoothly transitioning to constrained training. Simultaneously, cosine annealing learning rate scheduling is used, maintaining a large learning rate in the early stages of training to help escape local minima, and gradually reducing the learning rate in the later stages for fine-tuning convergence. The fine-tuning stage typically continues training for several hundred epochs, observing that the loss composition gradually shifts from being dominated by data error to being dominated by physical error, indicating that the model gradually satisfies various physical constraints while maintaining fitting accuracy.

[0030] (c) Fine-tuning stage: Finally, the L-BFGS second-order optimizer is used to fine-tune the model. In this stage, the weights of each constraint term are fixed, focusing on the precise search for the extreme value of the total loss. L-BFGS utilizes second-order gradient information to more fully explore the shape of the loss and find a better solution. However, its iteration overhead is relatively large. To ensure efficiency, this invention adopts measures such as limited memory and gradient truncation: limiting the length of historical iteration memory (e.g., 100 steps), setting an upper limit on the gradient norm to avoid instability caused by Hessian ill-conditioning, and using a line search that satisfies the Wolfe condition to determine the step size for each step. After dozens of iterations, the model basically converges, resulting in the final PINN model.

[0031] By employing the aforementioned phased training strategy, the potential conflict between simultaneously optimizing data fitting and physical constraints is successfully resolved, resulting in efficient and stable model training. Comparative experiments show that compared to one-time end-to-end training, this strategy reduces the mean absolute error on the test set by approximately 23.5% and the physical constraint violation rate by approximately 18.7%. Simultaneously, the model's robustness to initial parameters and random seeds is improved, with performance fluctuations under different initial conditions less than 2%. This demonstrates that the training scheme of this invention not only improves model accuracy but also ensures consistent and reliable training results.

[0032] Figure 5 Flowchart of the hybrid training strategy. The entire training process is divided into three phases, proceeding sequentially: pre-training (data supervision only), fine-tuning (adding physical constraints and using combined loss), and fine-tuning (using L-BFGS optimization to further improve accuracy). The arrow annotations indicate the conditions and objectives connecting each phase. Each phase employs different optimization strategies to gradually balance the model's accuracy and physical consistency, ultimately obtaining a high-accuracy model that satisfies physical laws.

[0033] Step (5) Structure optimization: Use the trained PINN model to perform structure optimization design.

[0034] After completing the training and fine-tuning of the PINN model, the core task of this step is to optimize the geometric parameters of the infrared anti-reflection micro / nano structure using its high-efficiency prediction capability. To simplify the optimization process, this invention adopts an intuitive grid search strategy. The method first discretizes the continuous parameters to be optimized, such as period T, height H, and duty cycle f, within the reasonable range determined in step (1) according to a preset fixed step size, thereby transforming the infinite continuous parameter space into a finite, multi-dimensional discrete parameter grid. Subsequently, the optimization program systematically traverses each node on the grid, and each node represents a specific combination of structural parameters (including discrete structure types such as square prisms or cylinders). For each combination in the grid, the system calls the pre-trained PINN model as an efficient proxy model to quickly calculate its average reflectivity within the preset target working band and incident angle range, and uses this as the performance evaluation value of the parameter combination. After completing the traversal and evaluation of all grid points, the node with the lowest performance evaluation value (i.e., the best anti-reflection effect) and its corresponding parameter combination can be locked by simple comparison. The final optimized design was selected and will be used directly to guide subsequent micro- and nano-fabrication and implementation.

[0035] Step (6) Structure fabrication: According to the optimized parameters, an anti-reflection micro-nano structure is fabricated on the surface of the infrared window using micro-nano fabrication technology.

[0036] This step aims to refine the optimal parameters determined in step (5). Microstructure machining is performed directly on the surface of the infrared window using ultra-precision single-point diamond turning (SPDT). This method requires the substrate material of the infrared window to be a diamond-machinable material, such as silicon (Si), germanium (Ge), or zinc sulfide (ZnS). During machining, the infrared window substrate is directly clamped onto the high-precision air-bearing spindle of an ultra-precision machine tool. Subsequently, based on the optimized cycle time...

[0037] and duty cycle The toolpath is generated using parameters, and a diamond tool is driven by a Fast Tool Servo (FTS) system to precisely remove material from the high-speed rotating window surface. The tool directly etchs micro- and nano-grooves into the substrate material through high-speed radial vibration at the micrometer scale, thus preserving the columnar array (square or cylindrical) in the design as uncut material; its height is determined by the depth of cut. Precise control is achieved through a meticulous cleaning process after turning, which thoroughly removes tiny debris and coolant generated during machining. This yields a finished infrared window with anti-reflective micro / nano structures directly integrated into its surface. Finally, the actual morphology and optical properties of the structure are measured using scanning electron microscopy (SEM) and spectral analysis to verify the high consistency between the precision of the direct machining and the design objectives.

[0038] Example: This embodiment uses a ZnS infrared window as an example, whose operating wavelength is 8~12μm. It is necessary to minimize surface reflectivity within this wavelength range. The implementation steps are as follows: (1) Parameter setting: A square column array microstructure was selected as the anti-reflection layer. The design parameter range was determined based on processing capability and theoretical analysis: period T was 0.5~5μm, height H was 0.1~3μm, square column duty cycle f (the ratio of column width to period) was 0~1, and working wavelength was... Range 8~12μm, incident angle Consider a uniform distribution within the range of 0–45°. This range covers the potential region of optimal anti-reflective structural parameters.

[0039] (2) Sample Acquisition: According to the parameter space sampling strategy of this invention, a training sample set is generated within the above range. For example, the period T is sampled according to importance, with a focus on small period segments, to obtain 50 representative values; the height H is sampled according to Latin hypercube to obtain several candidate values, and the duty cycle f of the square cylinder is refined in combination with the resonance condition; the working wavelength... Several points were uniformly selected within the range of 8–12 μm, and the incident angle was... Within the 0–45° range, the large-angle region is appropriately densified. The reflectivity of each set of structural parameters at various sampling wavelengths and incident angles is calculated using RCWA electromagnetic simulation to form training data. If the initial data is insufficient, mirror samples can be generated using reciprocity to expand the dataset.

[0040] (3) Model training: Construct the PINN network model, including the input layer ( The model consists of 5 hidden layers and an output layer with physical constraints, along with structure type encoding. Pre-training is performed first, using only simulation data to supervise the model's learning of basic relationships. Once the validation set error converges, physical constraints are introduced for fine-tuning: the energy conservation hard constraint module is enabled, and reciprocity symmetry penalty and resonant gradient penalty terms are added to the loss function. Training continues until the model achieves low data and physical errors simultaneously. Finally, L-BFGS optimization is used for fine-tuning to improve model accuracy. After training, a PINN model capable of quickly predicting reflectivity is obtained.

[0041] (4) Structure optimization: Use the trained PINN model to perform structure optimization design.

[0042] ZnS is a typical optical material for infrared windows. This step designs anti-reflective micro / nanostructures on the ZnS surface to minimize its average reflectivity in the mid-to-far infrared band (3μm to 14μm) and at incident angles of 0° to 30°. To achieve this goal, a grid search strategy is used to optimize the structural parameters. Specifically, the search range for the structural period T is set to [1.0μm, 3.0μm] with a discretization step size of 0.05μm; the search range for the height H is set to [1.0μm, 2.5μm] with a step size of 0.05μm; and the search range for the duty cycle f is set to [0.3, 0.7] with a step size of 0.02. Simultaneously, both structural types S (square column arrays and cylindrical arrays) are traversed. The optimization program uses the pre-trained PINN model to quickly calculate the average reflectivity within the target band and angle range for each node combination in the above parameter grid as a performance evaluation value. After completing the traversal and evaluation of all grid point combinations, this embodiment identifies the optimal parameter combination with the lowest performance evaluation value, namely: structure type. It is a square column array with a period of = 2.25μm, height = 1.80μm, duty cycle = 0.52. This parameter combination

[0043] It was selected as the final optimized design scheme to guide subsequent micro-nano fabrication and implementation.

[0044] (5) Structural implementation: based on the optimized parameter combination The microstructure was directly machined on a ZnS window substrate with a diameter of 50 mm and a thickness of 5 mm using the ultra-precision single-point diamond turning (SPDT) process.

[0045] First, the ZnS substrate was mounted on a high-precision air-bearing spindle of an ultra-precision machine tool. Then, a turning toolpath was generated based on optimal period and duty cycle parameters, and the turning depth was set to an optimal height of 1.80 μm. A diamond tool was controlled by a FastTool Servo (FTS) system to perform high-speed radial motion while the substrate rotated at high speed, precisely removing material and directly machining a square column array with a predetermined period onto the ZnS substrate. After turning, the workpiece was cleaned with deionized water to thoroughly remove micro-debris and coolant generated during machining, ultimately obtaining a finished infrared window with an anti-reflection micro / nano structure directly integrated on its surface. Finally, the actual morphology and optical properties of the structure were measured and characterized using a scanning electron microscope and an infrared spectrometer. The results showed that the processed structural parameters highly matched the design values, and its anti-reflection performance in the target wavelength band was excellent, verifying the closed-loop effectiveness of the method from design to implementation.

[0046] Because this invention takes into account processing constraints, the selected parameters are within the range of existing process capabilities, allowing for successful processing and shaping. Spectroscopic testing of the prepared samples revealed a significant increase in transmittance at 8–12 μm compared to the unprocessed substrate, demonstrating the effectiveness of the design method of this invention.

[0047] This embodiment demonstrates the application process of the method of the present invention in practical design. The results show that the square columnar microstructure designed by the present invention exhibits excellent anti-reflection performance, maintaining low reflectivity under broad spectral and multi-angle conditions. Moreover, the design process is significantly simplified compared to traditional methods: using the PINN model, a large number of schemes can be evaluated and the optimal structure locked in within seconds, greatly reducing the workload of repeated simulations. In addition to square columnar structures, the present invention is also applicable to the anti-reflection design of other types of micro / nano structures (such as cylindrical arrays, conical structures, etc.), requiring only corresponding adjustments to the structure type input. In summary, the present invention provides an efficient and universal approach to optimizing the design of infrared anti-reflection microstructures, bringing new technical means to improve the performance of infrared optical components.

Claims

1. A method for optimizing the design of anti-reflection micro / nano structures on infrared window surfaces based on PINN, characterized in that... The method includes the following steps: Step (1) Formulate a parameter space sampling strategy based on physical priors: Theoretical analysis and engineering constraints are used to determine the reasonable range of structural geometry type S, period T, height H, duty cycle f, working wavelength λ and incident angle θ, and a differentiated sampling method is adopted for each parameter to cover the key areas of the high-dimensional parameter space with a limited number of samples; Step (2) Construct the PINN model architecture that incorporates physical priors: The PINN model architecture is based on PINN and consists of an input layer, a hidden layer, and an output layer. Step (3) Embed physical constraints into the PINN model: To further ensure the physical reliability of the model predictions, multiple electromagnetic physical constraints, such as the law of conservation of energy, reciprocity symmetry, and wavelength-period resonance conditions, are explicitly integrated into the model training process. Step (4) Phased and gradual training: The training process is divided into three stages: pre-training, physical fine-tuning, and fine-tuning, and constraints are gradually introduced to balance model accuracy and physical consistency. Step (5) Structural optimization: Utilize the trained PINN model for structural optimization design; Step (6) Structure Implementation: Based on the optimized parameters, anti-reflection micro-nano structures were fabricated on the surface of the infrared window using micro-nano fabrication processes.

2. The method for optimizing the design of anti-reflection micro / nano structures for infrared window surfaces based on PINN according to claim 1, characterized in that... The specific steps of step (1) are as follows: (a) Period T: Based on the equivalent medium theory, the upper limit constraint of the period is determined. Considering the processing limitations, the actual period range is extended to [0.5~5μm]. Within this range, importance sampling is adopted, and an exponential decay weight function is introduced to make the samples distributed in the sensitive interval with high information density, ensuring sufficient sampling of key small period regions, while avoiding unnecessary sampling of large period regions. (b) Height H and duty cycle f: First, the theoretical optimal range of height H is determined by using the film resonance condition. Combined with the processing depth capability, the height is limited to 0.1~3μm. Within this range, height sample points are generated by Latin hypercube sampling. Then, for each height sample, the corresponding optimal duty cycle f is calculated according to the resonance condition formula, and the duty cycle is refined by additional dense sampling in its neighborhood, thus forming a non-uniform two-dimensional sampling grid around the theoretical resonance condition. (c) Wavelength λ: For the transmission band of the material, discrete wavelength points are obtained by uniform sampling. These wavelength sampling points uniformly cover the working band of the infrared window to ensure that the training data covers the typical response of the entire spectral range. (d) Angle of incidence θ: Angle samples are selected by uniformly distributing the cosine value of the angle of incidence within the range of [0°~60°].

3. The method for optimizing the design of anti-reflection micro / nano structures for infrared window surfaces based on PINN according to claim 1, characterized in that... In step (2), the input layer uses five continuous parameters—period T, height H, duty cycle f, wavelength λ, and incident angle θ—and one discrete structure type parameter G as input; the hidden layer is a five-layer fully connected structure, each layer containing 128 neurons, and introduces the Swish activation function, and introduces a wavelength-period attention mechanism in the third hidden layer; the output layer adopts a dual-channel structure to predict the reflectivity of S-polarization and P-polarization respectively, embeds an energy conservation hard constraint module at the output end, and uses a constructive projection method to force the output to strictly satisfy energy conservation.

4. The optimized design method for anti-reflection micro / nano structures on infrared window surfaces based on PINN according to claim 3, characterized in that... In the input layer, to accommodate multimodal inputs, continuous parameters are first normalized to a range of [0,1]. For the structure type parameter G, one-hot encoding is used to encode the columnar array into a two-dimensional vector of (1,0) or (0,1). Finally, the input layer maps the processed 6-dimensional feature vector to the hidden layer through a fully connected layer as the starting point for subsequent network computations.

5. The optimized design method for anti-reflection micro / nano structures on infrared window surfaces based on PINN according to claim 1, characterized in that... The specific steps of step (3) are as follows: (a) Hard constraint of energy conservation: Energy conservation is directly achieved through the network output layer; (b) Reciprocity symmetric constraint: On the one hand, during dataset preprocessing, for each original sample Add an extra "mirror" sample Pairs of data are input into the network for training; on the other hand, a symmetry penalty term is introduced into the loss function of the model training, that is, the square of the difference between the two predicted reflectances obtained from the mirror input is calculated and weighted and added to the total loss; (c) Resonance gradient soft constraint: For parameter region samples that theoretically have resonance, the gradient of their reflectance with respect to wavelength is calculated in advance. During training, in addition to minimizing the error between the predicted value and the true value, the difference between the gradient of the model prediction curve with respect to wavelength and the theoretical gradient is also minimized. To ensure the effectiveness of gradient penalty, automatic differentiation is used to obtain the derivative of the PINN output with respect to wavelength, and the gradient is subjected to Gaussian filtering for noise reduction and smoothing.

6. The method for optimizing the design of anti-reflection micro / nano structures for infrared window surfaces based on PINN according to claim 1, characterized in that... The specific steps of step (4) are as follows: (a) Pre-training phase: The network is pre-trained using a purely data-driven approach; (b) Physical fine-tuning stage: After obtaining the pre-trained model, physical constraints are gradually added to fine-tune the model; (c) Fine-tuning stage: The L-BFGS second-order optimizer is used to fine-tune the model to obtain the final PINN model.

7. The method for optimizing the design of anti-reflection micro / nano structures for infrared window surfaces based on PINN according to claim 1, characterized in that... The specific steps of step (5) are as follows: (a) Discretize the continuous parameters that need to be optimized within the reasonable range of values ​​determined in step (1) according to a preset fixed step size, thereby transforming the infinite continuous parameter space into a finite, multidimensional discrete parameter grid. (b) Traverse each node on the grid. Each node represents a specific combination of structural parameters. For each combination in the grid, use the trained PINN model as an efficient surrogate model to quickly calculate its average reflectivity within the preset target working band and incident angle range, and use this as the performance evaluation value of the parameter combination. (c) After completing the traversal and evaluation of all grid points, the node with the lowest performance evaluation value is identified by comparison, and its corresponding parameter combination is determined. The final optimized design was selected and will be used directly to guide subsequent micro- and nano-fabrication and implementation.

8. The method for optimizing the design of anti-reflection micro / nano structures for infrared window surfaces based on PINN according to claim 1, characterized in that... The specific steps of step (6) are as follows: (a) The infrared window substrate is directly clamped onto the high-precision air-bearing spindle of an ultra-precision machine tool; (b) Based on the optimal parameters determined in step (5), a tool path is generated, and a diamond tool is driven by a fast tool servo system to precisely remove material from the high-speed rotating window surface; (c) After turning, the tiny debris and coolant generated during the machining process are thoroughly removed through a cleaning process to obtain a finished infrared window with anti-reflective micro-nano structures directly integrated on the surface.

9. The method for optimizing the design of anti-reflection micro / nano structures for infrared window surfaces based on PINN according to claim 8, characterized in that... The base material of the infrared window is a machinable diamond material.

10. The method for optimizing the design of anti-reflection micro / nano structures for infrared window surfaces based on PINN according to claim 9, characterized in that... The diamond-machinable material is one of silicon, germanium, or zinc sulfide (ZnS).

Images