The design of freeform progressive addition lenses is to provide wearers with a seamless transition between distance and near vision. As human lifespan increases and digital lifestyles become more prevalent, progressive lenses have become an essential visual aid. Although research on progressive lenses in China has advanced and design techniques have become more sophisticated, existing design methods still have certain limitations. For example, the computation of sagittal height is resource-intensive, with high time costs and limited precision. These limitations can cause the addition power to fail to meet expected standards, reduce the effective visual area, and make wearers feel uncomfortable when switching between distance and near vision. This, in turn, increases the visual adaptation time and may prevent the initial lens design from fully achieving its intended purpose. The goal of our study is to explore a more efficient and precise solution by proposing a method based on physics-informed neural network (PINN) for solving nonlinear partial differential equations and applying it to optimize the design of progressive addition lenses. Traditional numerical methods for solving nonlinear partial differential equations often face challenges such as high computational complexity and slow convergence rates. This innovative computational model optimizes the sagittal height distribution by minimizing the error between the neural network output and the governing equations, breaks the limitations of dimensionality, avoids truncation errors and numerical integration errors of variational forms, and overcomes the constraints of traditional design methods. This enables precise simulation of the sagittal height of progressive addition lenses, thereby improving lens optical performance and enhancing user visual experience.

We describe the theoretical foundation of the partial differential equations (PDEs) guiding the design of progressive addition lenses and propose a method for solving the PDEs of progressive addition lenses using a PINN. This method combines a neural network with the physical optical model of the lenses. First, we build a fully convolutional network model. We apply automatic differentiation to the sagittal height matrix output by the network in different directions to construct a loss function that quantifies the residuals of the nonlinear PDEs for progressive addition lenses. Additionally, we impose constraints by calculating the residuals after applying boundary conditions. Finally, during the iterative process, we minimize the loss functions under multiple constraints to update the neural network parameters (weights and biases). When the network converges, this method outputs an optimized sagittal height distribution, enabling precise simulation of the sagittal height of progressive addition lenses. Compared to traditional data-driven deep learning methods, this approach reduces the reliance on large amounts of training data and provides a better understanding of the physical processes, enhancing the interpretability of the neural network. In contrast to conventional numerical methods for solving PDEs, this method improves the optical performance of the lenses. Under the same lens design parameters, we design three lenses with different diopters using both numerical methods and the PINN approach, resulting in six sets of lens surface profiles. Then we manufacture and test them. We analyze the effect of this optimization method on the optical performance of progressive multifocal lenses, including optical power and astigmatism.

In our study, we design six sets of progressive addition lenses with different diopters using both numerical methods and the PINN approach. We develop a simulation program based on the differential geometry evaluation formulas for progressive multifocal lenses and, in combination with FFV simulation software, calculate the simulated optical power and astigmatism distribution data for the six sets of lenses. These results are used to evaluate the simulation outcomes and the optical performance of the lenses. Subsequently, we manufacture the designed lenses into physical lenses. We conduct detailed measurements of the actual lenses using a surface profilometer to obtain the actual optical power and astigmatism distribution data. Based on this, we use CAD software to measure and mark the angles and corridor widths of the distance and near vision zones of the lenses, aiming to assess the practical optimization effectiveness of the proposed algorithm and analyze the actual optical performance of the lenses. The difference between the sagittal height distributions predicted by the PINN and those obtained using the explicit finite difference method is approximately 1 μm. This result not only validates the predictive accuracy of the PINN model but also demonstrates the strong approximation capability and excellent learning ability of the proposed PINN model (Fig. 4). By comparing the six sets of lenses designed using both numerical methods and the PINN approach, it is evident that lenses 4 to 6 exhibit varying degrees of optimization compared to lenses 1 to 3. The optical power error in the distance vision area of lenses 4 to 6, relative to the preset theoretical values, is controlled within 0.05 D, with a significant reduction in astigmatism in the distance vision area. Additionally, the addition power (ADD) value more closely aligns with the preset theoretical value of 2.00 D (Tables 3?5). While optimizing optical performance, the effective visual area of both the distance and near vision zones remains largely consistent, with lens 5 outperforming lens 3 (Figs. 5?7). The manufacturing results closely match the simulation results, indicating that the PINN method effectively enhances the optical performance of progressive multifocal freeform lenses (Table 6).

Compared to traditional data-driven deep learning methods, the proposed approach reduces reliance on large amounts of training data and provides a better understanding of physical processes, enhancing the interpretability of the neural network. In contrast to traditional numerical methods for solving partial differential equations, it improves the optical performance of the lenses. In this study, we design six sets of progressive addition lenses with different diopters using both numerical methods and the PINN approach. Comparative analysis shows that the PINN-based method for solving partial differential equations effectively optimizes the design of progressive addition lenses and improves their optical performance to some extent.