arXiv subjects

Vasiliy A. Es'kin

Publications and source records attributed to Vasiliy A. Es'kin.

11 recordsLinked to original sources

arXiv2026

A Two-Mirror Faceted Projection System for EUV Lithography

We propose an all-reflective two-mirror projection system for extreme ultraviolet (EUV) lithography operating at exposure wavelengths of $13.5$~nm (Mo/Si) and $11.2$~nm (Ru/Be), delivering a fourfold ($4\times$) demagnification of the periodic mask pattern at a numerical aperture approaching unity ($\mathrm{NA}_{\max} \approx 0.993$). In contrast to conventional EUV projection objectives that incorporate 6--10 aspheric mirrors with an overall optical throughput of less than $15\%$, the proposed design redirects each accepted discrete spatial diffraction order scattered by the mask onto the wafer via a dedicated pair of planar mirror facets. The number of reflections is strictly fixed at two for all accepted orders, retaining $50$--$60\%$ of the power leaving the mask in each accepted order. We derive a spatial geometry providing rigorous optical path length equalization across all diffraction orders, thereby removing order-dependent propagation phase shifts. Individually optimized 30-bilayer Bragg multilayer coatings are designed for each facet using the transfer matrix method combined with global evolutionary optimization algorithms. The architecture is generalized to a three-dimensional vector formulation with a two-dimensionally periodic mask. Utilizing inverse lithography technology, Fourier parameterization, and a differentiable electromagnetic modal waveguide solver, we solve the synthesis problem for binary absorber masks (La absorber on a Ru/Be/Sr multilayer mirror). We demonstrate simulated aerial images of sub-10-nm features on the wafer (isolated peaks with a full width at half maximum (FWHM) of approximately $5.4$~nm and line pairs with a critical dimension of $6$~nm) and find that the two peaks remain resolved for the tested wafer defocus values from $0$ to $5$~nm along the $z$-axis.

physics.optics↗

arXiv2026

Gradient-based inverse lithography for EUV masks via the waveguide method and a physics-informed neural operator

Gradient-based inverse lithography technology~(ILT) for extreme ultraviolet~(EUV) masks is presented. A novel framework treats the differentiable waveguide method and the recently proposed waveguide neural operator~(WGNO) as end-to-end physics engines, recovering the permittivity of the absorber of the mask through automatic differentiation of the full forward diffraction model. Numerical experiments on realistic 2D and 3D absorbers of the mask (TaBN, La, U) at $λ{=}11.2$~nm show that the considered ILT methods make it possible to obtain a mask structure that achieves the desired field on the wafer.

cs.LG↗

arXiv2026

Dynamical Systems Theory Behind a Hierarchical Reasoning Model

Current large language models (LLMs) primarily rely on linear sequence generation and massive parameter counts, yet they severely struggle with complex algorithmic reasoning. While recent reasoning architectures, such as the Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), demonstrate that compact recursive networks can tackle these tasks, their training dynamics often lack rigorous mathematical guarantees, leading to instability and representational collapse. We propose the Contraction Mapping Model (CMM), a novel architecture that reformulates discrete recursive reasoning into continuous Neural Ordinary and Stochastic Differential Equations (NODEs/NSDEs). By explicitly enforcing the convergence of the latent phase point to a stable equilibrium state and mitigating feature collapse with a hyperspherical repulsion loss, the CMM provides a mathematically grounded and highly stable reasoning engine. On the Sudoku-Extreme benchmark, a 5M-parameter CMM achieves a state-of-the-art accuracy of 93.7 %, outperforming the 27M-parameter HRM (55.0 %) and 5M-parameter TRM (87.4 %). Remarkably, even when aggressively compressed to an ultra-tiny footprint of just 0.26M parameters, the CMM retains robust predictive power, achieving 85.4 % on Sudoku-Extreme and 82.2 % on the Maze benchmark. These results establish a new frontier for extreme parameter efficiency, proving that mathematically rigorous latent dynamics can effectively replace brute-force scaling in artificial reasoning.

cs.AI↗

arXiv2026

Physics-Informed Neural Systems for the Simulation of EUV Electromagnetic Wave Diffraction from a Lithography Mask

Physics-informed neural networks (PINNs) and neural operators (NOs) for solving the problem of diffraction of Extreme Ultraviolet (EUV) electromagnetic waves from contemporary lithography masks are presented. A novel hybrid Waveguide Neural Operator (WGNO) is introduced, based on a waveguide method with its most computationally expensive components replaced by a neural network. To evaluate performance, the accuracy and inference time of PINNs and NOs are compared against modern numerical solvers for a series of problems with known exact solutions. The emphasis is placed on investigation of solution accuracy by considered artificial neural systems for 13.5 nm and 11.2 nm wavelengths. Numerical experiments on realistic 2D and 3D masks demonstrate that PINNs and neural operators achieve competitive accuracy and significantly reduced prediction times, with the proposed WGNO architecture reaching state-of-the-art performance. The presented neural operator has pronounced generalizing properties, meaning that for unseen problem parameters it delivers a solution accuracy close to that for parameters seen in the training dataset. These results provide a highly efficient solution for accelerating the design and optimization workflows of next-generation lithography masks.

cs.LG↗

arXiv2025

Physics-informed neural networks and neural operators for a study of EUV electromagnetic wave diffraction from a lithography mask

Physics-informed neural networks (PINNs) and neural operators (NOs) for solving the problem of diffraction of Extreme Ultraviolet (EUV) electromagnetic waves from a mask are presented. A novel hybrid Waveguide Neural Operator (WGNO) is introduced, which is based on a waveguide method with its most computationally expensive part replaced by a neural network. Numerical experiments on realistic 2D and 3D masks show that the WGNO achieves state-of-the-art accuracy and inference time, providing a highly efficient solution for accelerating the design workflows of lithography masks.

math.NA↗

arXiv2024

About rectified sigmoid function for enhancing the accuracy of Physics-Informed Neural Networks

The article is devoted to the study of neural networks with one hidden layer and a modified activation function for solving physical problems. A rectified sigmoid activation function has been proposed to solve physical problems described by the ODE with neural networks. Algorithms for physics-informed data-driven initialization of a neural network and a neuron-by-neuron gradient-free fitting method have been presented for the neural network with this activation function. Numerical experiments demonstrate the superiority of neural networks with a rectified sigmoid function over neural networks with a sigmoid function in the accuracy of solving physical problems (harmonic oscillator, relativistic slingshot, and Lorentz system).

math.NA↗

arXiv2024

Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?

The article discusses the development of various methods and techniques for initializing and training neural networks with a single hidden layer, as well as training a separable physics-informed neural network consisting of neural networks with a single hidden layer to solve physical problems described by ordinary differential equations (ODEs) and partial differential equations (PDEs). A method for strictly deterministic initialization of a neural network with one hidden layer for solving physical problems described by an ODE is proposed. Modifications to existing methods for weighting the loss function are given, as well as new methods developed for training strictly deterministic-initialized neural networks to solve ODEs (detaching, additional weighting based on the second derivative, predicted solution-based weighting, relative residuals). An algorithm for physics-informed data-driven initialization of a neural network with one hidden layer is proposed. A neural network with pronounced generalizing properties is presented, whose generalizing abilities of which can be precisely controlled by adjusting network parameters. A metric for measuring the generalization of such neural network has been introduced. A gradient-free neuron-by-neuron fitting method has been developed for adjusting the parameters of a single-hidden-layer neural network, which does not require the use of an optimizer or solver for its implementation. The proposed methods have been extended to 2D problems using the separable physics-informed neural networks approach. Numerous experiments have been carried out to develop the above methods and approaches. Experiments on physical problems, such as solving various ODEs and PDEs, have demonstrated that these methods for initializing and training neural networks with one or two hidden layers (SPINN) achieve competitive accuracy and, in some cases, state-of-the-art results.

math.NA↗

arXiv2024

Separable Physics-Informed Neural Networks for the solution of elasticity problems

A method for solving elasticity problems based on separable physics-informed neural networks (SPINN) in conjunction with the deep energy method (DEM) is presented. Numerical experiments have been carried out for a number of problems showing that this method has a significantly higher convergence rate and accuracy than the vanilla physics-informed neural networks (PINN) and even SPINN based on a system of partial differential equations (PDEs). In addition, using the SPINN in the framework of DEM approach it is possible to solve problems of the linear theory of elasticity on complex geometries, which is unachievable with the help of PINNs in frames of partial differential equations. Considered problems are very close to the industrial problems in terms of geometry, loading, and material parameters.

math.NA↗

arXiv2024

Catenary and Mercator projection

The Mercator projection is sometimes confused with another mapping technique, specifically the central cylindrical projection, which projects the Earth's surface onto a cylinder tangent to the equator, as if a light source is at the Earth's center. Accidentally, this misconception is rather close to a truth. The only operation that the map needs is a free bending in a uniform gravitational field if the map's material is dense and soft enough to produce a catenary profile. The north and south edges of the map should be parallel and placed in the same plane at the appropriate distance. In this case, the bent map been projected onto this plane gives the Mercator projection. This property is rather curious, since it allows to make such a sophisticated one-to-one mapping as the Mercator projection using simple tools available in the workroom.

math.NA↗

arXiv2023

About optimal loss function for training physics-informed neural networks under respecting causality

A method is presented that allows to reduce a problem described by differential equations with initial and boundary conditions to the problem described only by differential equations. The advantage of using the modified problem for physics-informed neural networks (PINNs) methodology is that it becomes possible to represent the loss function in the form of a single term associated with differential equations, thus eliminating the need to tune the scaling coefficients for the terms related to boundary and initial conditions. The weighted loss functions respecting causality were modified and new weighted loss functions based on generalized functions are derived. Numerical experiments have been carried out for a number of problems, demonstrating the accuracy of the proposed methods.

math.NA↗

arXiv2012

Multiple Scattering of Electromagnetic Waves by an Array of Parallel Gyrotropic Rods

We study multiple scattering of electromagnetic waves by an array of parallel gyrotropic circular rods and show that such an array can exhibit fairly unusual scattering properties and provide, under certain conditions, a giant enhancement of the scattered field. Among the scattering patterns of such an array at its resonant frequencies, the most amazing is the distribution of the total field in the form of a perfect self-similar structure of chessboard type. The scattering characteristics of the array are found to be essentially determined by the resonant properties of its gyrotropic elements and cannot be realized for arrays of nongyrotropic rods. It is expected that the results obtained can lead to a wide variety of practical applications.

physics.class-ph↗