Operator-Consistent Physics-Informed Learning for Wafer Thermal Reconstruction in Lithography

TL;DR

The paper addresses accurate thermal field reconstruction in post-exposure bake where traditional PINNs struggle due to misalignment of geometry, fields, and operators. It proposes an operator-aligned LSTM-gated Liquid Neural Network architecture that unifies coordinates, field variables, and differential operators in a single computation graph, with an energy-based weak formulation and operator-residual losses. Key contributions include decoupled gating for stable conditioning, a single-step preconditioned residual block, and an explicit operator-aligned objective validated on a 2D circular wafer with Robin boundaries; the LSTM-LNN-PINN achieves RMSE of 3.83e-5 and uniform errors under 1e-4, outperforming baselines. This approach yields stable, accurate surrogates for wafer-scale PEB thermal analysis and offers a transferable strategy for operator-consistent neural surrogates in other physical domains.

Abstract

Thermal field reconstruction in post-exposure bake (PEB) is critical for advanced lithography, yet current physics-informed neural networks (PINNs) suffer from inconsistent accuracy due to a misalignment between geometric coordinates, physical fields, and differential operators. To resolve this, we introduce a novel architecture that unifies these elements on a single computation graph by integrating LSTM-gated mechanisms within a Liquid Neural Network (LNN) backbone. This specific combination of gated liquid layers is necessary to dynamically regulate the network's spectral behavior and enforce operator-level consistency, which ensures stable training and high-fidelity predictions. Applied to a 2D PEB scenario with internal heat generation and convective boundaries, our model formulates residuals via differential forms and a composite loss functional. The results demonstrate rapid convergence, uniformly low errors, strong agreement with FEM benchmarks, and stable training without late-stage oscillations, outperforming existing baselines in accuracy and robustness. Our framework thus establishes a reliable foundation for high-fidelity thermal modeling and offers a transferable strategy for operator-consistent neural surrogates in other physical domains.

Figures (5)

• Figure 1: We present an LSTM-gated LNN-PINN architecture that maps spatial coordinates to temperature via a shared trunk and branches into physics heads for PDE, boundary/port, and source residuals. Residual channels aggregate into a single energy-consistent loss, stabilizing training and delivering high-fidelity predictions with accurate boundary traces. (a) Structure of the Physics-Informed Neural Network (LSTM-LNN-PINN) incorporating Gated Liquid Neural Networks. The original Multi-Layer Perceptron (MLP) is replaced by Liquid Neural Network (LNN) modules with Long Short-Term Memory (LSTM) gating mechanisms, forming a two-layer gated LNN network with 64 neurons per layer. This design dynamically adjusts information flow and feature extraction capabilities, enhancing modeling performance for spatiotemporal nonlinear behaviors in complex PEB process heat transfer fields while preserving physical constraints. The figure illustrates the interconnections and data flow among submodules, reflecting the deep integration of physical knowledge with neural network innovations. (b) Computation diagram of a Long Short-Term Memory (LSTM) unit, showing the input gate, forget gate, output gate, and candidate cell state. The figure illustrates affine transformations of weights and biases, nonlinear activation mappings, element-wise gate multiplications, weighted updates of previous cell state $\mathbf{c}_{0}$, generation of current cell state $\mathbf{c}$, and final hidden state $\mathbf{h}$ modulated by $\tanh$ and output gate. This demonstrates the memory retention and selective forgetting mechanisms inherent to Long Short-Term Memory (LSTM) units for sequence modeling in PEB process heat transfer fields while gates act as input-conditioned diagonal preconditioners along operator-aligned axes.
• Figure 2: Temperature distribution obtained by numerically simulating the PEB process heat transfer problem using the finite element method on the MATLAB platform. This result serves as a reference benchmark for subsequent computation of RMSE and absolute error distributions for different model predictions. The figure illustrates the spatial distribution of the temperature field within the computational domain, reflecting the effects of finite element discretization and boundary conditions on PEB process thermal behavior. (a) We depict the post-exposure bake (PEB) setup: a circular wafer domain $\Omega$ (radius $R$) with internal heat generation $Q$ and a convective (Robin) boundary characterized by coefficient $h$ and ambient temperature $T_\infty$; arrows indicate heat flux and boundary exchange. (b) We show the finite-element (FEM) mesh that provides the reference solution, i.e., a conforming triangular tessellation of the circular domain with characteristic element size $H_{\max}$, respecting the geometry and boundary ports. (c) We present representative outcomes under this setup: the steady-state temperature field $T(x,y)$ and the absolute-error map $\lvert T_{\text{model}}(x,y)-T_{\text{FEM}}(x,y)\rvert$, which we use for visual diagnostics and RMSE evaluation.
• Figure 3: (a) RMSE versus learning rate (1e-4–1e-3; step 1e-4) for PINN, LNN-PINN, LSTM-PINN, and LSTM-LNN-PINN, highlighting robustness, sensitivity, and best-performing rates. (b) Absolute-error maps relative to the FEM reference across the same rates, revealing spatial patterns and magnitudes of model errors. (c) Predicted steady-state temperature fields across the same rates.
• Figure 4: Training loss curves of PINN, LNN-PINN, LSTM-PINN and LSTM-LNN-PINN models under multiple learning rate settings plotted in linear coordinates. Learning rates start from 1e-4 and increase by 1e-4 up to 1e-3, with each curve representing the dynamic evolution of training loss for a specific learning rate. This figure reflects the convergence trends of the model during training and provides a visual basis for understanding the performance of the LSTM-LNN structure within the PINN framework and the effects of learning rate adjustments.
• Figure 5: Training loss curves of PINN, LNN-PINN, LSTM-PINN and LSTM-LNN-PINN models under different learning rate settings plotted in logarithmic coordinates. Learning rates range from 1e-4 to 1e-3 with increments of 1e-4. The horizontal axis represents training iterations, and the vertical axis is the logarithmic scale of the loss function, highlighting magnitude changes and subtle fluctuations under different learning rates. This log-scale loss plot provides an important visual reference for evaluating the convergence performance and training stability of the LSTM-LNN-PINN model.