Abstract. Background: Modern one-digit technological nodes demand strict reproduction of the optical proximity corrections for repeatable congruent patterns. To ensure this property, the optical and process simulations must be invariant to the geometrical transformations of the translation, rotation, and reflection. Simulators must support invariance both in theory, mathematically, and in practice, numerically. The invariance of compact modeling operators has never been scrutinized before. Aim: We aim to examine manner and conditions under which optical simulations preserve or violate intrinsic invariances of exact imaging. We analyze invariances of Volterra operators, which are widely used in compact process modeling. Our goal is to determine necessary and sufficient conditions under which such operators become fully invariant Approach: We use theoretical analysis to deduce full invariance conditions and numerical simulations to illustrate the results. Results: The linear fully invariant operators are convolutions with rotationally symmetrical kernels. The fully invariant quadratic operators have special functional form with two radial and one polar argument and are not necessarily rotationally symmetrical. We deduced invariance conditions for the kernels of high-order Volterra operators. Conclusions: We suggest to use fully invariant nonlinear Volterra operators as atomic construction blocks in machine learning and neural networks for compact process modeling.

中文翻译：

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紧凑过程建模中的变换不变性

摘要。背景：现代一位数技术节点要求严格复制可重复全等模式的光学邻近校正。为了确保这一特性，光学和过程模拟必须对平移、旋转和反射的几何变换保持不变。模拟器必须在理论上、数学上和在实践中在数值上支持不变性。紧凑建模算子的不变性以前从未被仔细研究过。目的：我们的目标是检查光学模拟保持或违反精确成像的内在不变性的方式和条件。我们分析了广泛用于紧凑过程建模的 Volterra 算子的不变性。我们的目标是确定这些运算符成为完全不变的充分必要条件 方法：我们使用理论分析来推导出完全不变的条件和数值模拟来说明结果。结果：线性完全不变算子是具有旋转对称核的卷积。完全不变的二次算子具有特殊的函数形式，具有两个径向和一个极坐标，并且不一定是旋转对称的。我们推导出了高阶 Volterra 算子核的不变性条件。结论：我们建议使用完全不变的非线性 Volterra 算子作为机器学习和神经网络中的原子构建块，以进行紧凑的过程建模。完全不变的二次算子具有特殊的函数形式，具有两个径向和一个极坐标，并且不一定是旋转对称的。我们推导出了高阶 Volterra 算子核的不变性条件。结论：我们建议使用完全不变的非线性 Volterra 算子作为机器学习和神经网络中的原子构建块，以进行紧凑的过程建模。完全不变的二次算子具有特殊的函数形式，具有两个径向和一个极坐标，并且不一定是旋转对称的。我们推导出了高阶 Volterra 算子核的不变性条件。结论：我们建议使用完全不变的非线性 Volterra 算子作为机器学习和神经网络中的原子构建块，以进行紧凑的过程建模。