We present a global-in-time spectral Duhamel-Picard method with polynomial expansions for post-exposure bake (PEB) simulation. The coupled acid-quencher reaction-diffusion equations are reformulated as nonlinear Volterra integral equations using the Duhamel principle. Full trajectories are represented by finite polynomial expansions, with temporal coefficient fields determined over the bake interval by Anderson-accelerated Picard iteration, thereby avoiding sequential time marching. In 3-D PEB simulations, we employ Chebyshev and Legendre temporal bases while evaluating the Duhamel integrals using the corresponding Gaussian quadrature rules. We demonstrate that the proposed method simulates a large domain of 1200×1200×100 voxels on the order of one minute. The representation is further incorporated into a physics-informed neural operator (PINO) with a mixed-transform variant of a factorized Fourier neural operator, which is trained without precomputed solution trajectories. Together, the integral formulation and its PINO extension enable accelerated, time-marching-free prediction of concentration fields at arbitrary times within the bake interval.
Figures & tables
Gauss–Chebyshev
Gauss–Legendre
i
ξi
si(tPEB)[s]
wi(tPEB)[s]
ξi
si(tPEB)[s]
wi(tPEB)[s]
1
+0.980785
99.039
03.831
+0.960290
98.014
05.061
2
+0.831470
91.574
10.909
+0.796666
89.833
11.119
3
+0.555570
77.779
16.326
+0.525532
76.277
15.685
4
+0.195090
59.755
19.258
+0.183435
59.172
18.134
5
−0.195090
40.245
19.258
−0.183435
40.828
18.134
Table 1: Comparison of N=8 Gauss–Chebyshev and Gauss–Legendre quadrature nodes, physical times, and scaled weights over the full bake interval [0,tPEB] for tPEB=100s .
Figure 1: Random binary mask containing horizontally oriented features, used to construct the initial acid distribution in the 800×800×200 PEB benchmark domain.
Parameter
Description
Value
A0
Inside / outside mask
0.5 / 0.0
Q0
Uniform initial quencher concentration
0.05
P0
Uniform initial protected-group concentration
1.0
D0
Uniform initial deprotected-group concentration
0.0
k1,k4
Reaction rate constants
0.01–1
k2,k3,k5
Reaction rate constants
0.0001–0.01
Table 2: Fixed initial conditions and parameter ranges used in the temporal fitting benchmark.
Figure 2: Reference time-marching trajectories for acid ( A ) and protected group ( P ) over t=0 to 100 s, with representative acid diffusion coefficients Dacid=1 and 10nm2/s .
Figure 3: Representative temporal fitting of acid trajectories at sampled voxel locations selected to cover diverse local profile shapes.
Figure 4: Global fitting statistics over all voxel locations for the A , Q , P , and D fields: (a) coefficient of determination R2 , (b) trajectory mismatch relative to the time-marching reference, and (c) normalized coefficient contributions for polynomial expansions with different orders K .
Figure 5: Global-in-time spectral Duhamel–Picard solutions for the acid ( A ), quencher ( Q ), protected-group ( P ), and deprotected-group ( D ) fields at tPEB=50s for representative diffusion settings with Dacid=Dq∈{1,10}nm2/s . The corresponding absolute differences relative to the FD4–RK4 time-marching reference obtained using Δt=0.01s are shown alongside the computed fields.
Figure 6: Convergence behavior of the global-in-time spectral Duhamel–Picard solver for 100 parameter combinations generated by Latin hypercube sampling over the ranges listed in Table 2 . (a) Convergence histories of the sampled cases. (b) Relationship between k4 and the required number of Picard iterations.
Method
Device
Runtime (s)
T/TPicard
FD4–RK4
CPU
221,600
2.8×103
FD4–RK4
GPU
2,700
33.8
Global-in-time Spectral Duhamel-Picard
GPU
80
1
Table 3: Wall-clock runtime comparison between the global-in-time method and conventional FD4–RK4 method run on CPU and GPU.
Figure 7: Performance of the Physics-Informed Neural Operator (PINO) trained with the global-in-time PEB formulation evaluated for arbitrary validation masks at random bake times, which were not explicitly included in the training dataset. The figure shows the predicted fields and their differences relative to the reference PEB solution.
Figure 8: Mean absolute errors for (a) acid and (b) protected groups as a function of bake time.
Figure 9: Comparison of post-development resist profiles predicted by the global-in-time Duhamel–Picard method and the PINO surrogate for randomly sampled mask layouts and PEB times. (a) Three-dimensional resist profiles after development at tPEB=35 , 45 , and 86s . (b) Corresponding lateral contour comparisons at z=75 , 50 , and 25nm , with the Duhamel–Picard results shown in blue and the PINO predictions in red.
We present a physics-informed neural operator (PINO) trained with pseudo-spectral frequency-domain (PSFD) equations for electromagnetic (EM) scattering problems in EUV lithography. The Fourier neural operator is factorized into a two-dimensional lateral (xy) branch and a one-dimensional axial (z) branch and is trained self-consistently with background decomposition.Thus, the full-vector coupling between the mask and the multilayer response is retained without invoking a finite-order Born approximation. In this way, the computational domain size is significantly reduced, thereby lowering the computational cost. The PINO is trained on approximately 16,000 mask designs from the LithoBench library sampled randomly at each training iteration without using precomputed EM field solutions. The PINO surrogate model yields predictions with a mean absolute error of about 7×10−3 for the scattered intensity of held-out mask patterns relative to the reference PSFD solution. Combined with spectral damping, the PINO warm-start initialization accelerates the background-decomposed PSFD solver on finer discretizations.
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.
Vasiliy A. Es'kin, Egor V. Ivanov
Department of Radiophysics, University of Nizhny Novgorod 23 Gagarin Ave., Nizhny Novgorod 603022, Russia
Neuro-spectral architectures (NeuSA) are a class of physics-informed neural networks (PINNs) that have proven effective for solving time-dependent PDEs in the frequency domain. However, training and inference often rely on costly numerical integration of the learned vector field, typically using sequential Runge-Kutta time stepping, which limits temporal parallelism. We introduce PaPi, an integration scheme inspired by weakly nonlinear flows that enables time-Parallel inference through exponential integration and a Duhamel-Picard scheme. We show that the Duhamel-Picard iteration converges exponentially for all flows, and provide explicit error bounds that depend on how close the flow is to linear. Experiments show that PaPi accelerates training and inference by up to two orders of magnitude, achieving training times of under one second for low-dimensional PDE learning problems, while maintaining competitive predictive accuracy.
Arthur Bizzi, Márcio Marques, Kleyton Da Costa +4
Imperial College London London, United Kingdom · Instituto de Matemática Pura e Aplicada Rio de Janeiro, RJ, Brazil · Computer Science University College London London, United Kingdom +3