Deep learning–based inverse method for layout design

Abstract

Layout design is encountered in many fields of engineering and science. Those with complex constraints are particularly challenging to solve due to the non-uniqueness of the solution and the difficulties in incorporating the constraints into the conventional optimization-based methods. In this paper, we propose a design method based on the recently developed machine learning technique, variational autoencoder (VAE). We utilize the learning capability of the VAE to learn the constraints and the generative capability of the VAE to generate design candidates that automatically satisfy all the constraints. As such, no constraints need to be imposed during the design stage. In addition, we show that the VAE network is also capable of learning the underlying physics of the design problem, leading to an efficient design tool that does not need any physical simulation once the network is constructed. We demonstrated the performance of the method on two cases: inverse design of surface diffusion–induced morphology change and mask design for optical microlithography.

Surface diffusion solver

A phase-field model is used to describe interface evolution caused by surface diffusion, which is shown as follows:

$$ \frac{\partial \phi }{\partial t}=\nabla \bullet \frac{9}{4\varepsilon }M\left(\phi \right)\nabla \mu $$

(8)

$$ \mu =-\varepsilon \Delta \phi +{f}^{\prime}\left(\phi \right) $$

(9)

$$ f\left(\phi \right)=\frac{1}{4}{\left(1-{\phi}^2\right)}^2 $$

(10)

$$ M\left(\phi \right)={\left(1-{\phi}^2\right)}^2 $$

(11)

In these equations, ϕ denotes the order parameter for the phases of the system: ϕ =  − 1 represents a void phase and ϕ = 1 represents a solid phase; μ is the chemical potential; ε is a parameter controlling the thickness of the interface; f(ϕ) is the bulk free energy; and M(ϕ) is the mobility. As ε → 0, this model converges to Mullins sharp interface model (7) for surface diffusion, see (Lee et al. 2016).

The phase-field equations are solved numerically on a periodic domain by using an operator-splitting-based, quasi-spectral, semi-implicit time-stepping scheme. The semi-implicit scheme for (8) can be written in the following form (Jiang et al. 2012),

$$ {\left.\frac{\phi^{n+1}-{\phi}^n}{\Delta t}=\nabla \bullet \frac{9}{4\varepsilon }M\left(\phi \right)\nabla \mu \right|}_n-B{\Delta }^2\left({\phi}^{n+1}-{\phi}^n\right)+S\Delta \left({\phi}^{n+1}-{\phi}^n\right) $$

(12)

where B and S are two positive stabilizing parameters chosen to guarantee the numerical stability. Performing Fourier transform on (12), we obtain,

$$ \frac{{\widehat{\phi}}^{n+1}-{\widehat{\phi}}^n}{\Delta t}=F\left\{{\left.\nabla \bullet \frac{9}{4\varepsilon }M\left(\phi \right)\nabla \mu \right|}_n\right\}-B{\left({\left|k\right|}^2\right)}^2\left({\widehat{\phi}}^{n+1}-{\widehat{\phi}}^n\right)+S{\left|k\right|}^2\left({\widehat{\phi}}^{n+1}-{\widehat{\phi}}^n\right) $$

(13)

where $\widehat{\phi}(k)$ is the Fourier transform of ϕ(x), and F{∙} stands for Fourier transform.

Then, we can derive the following explicit time-stepping scheme in the spectrum space,

$$ {\widehat{\phi}}^{n+1}={\widehat{\phi}}^n+\frac{1}{1+\left(S{\left|k\right|}^2+B{\left({\left|k\right|}^2\right)}^2\right)\Delta t}\left(F\left\{{\left.\nabla \bullet \frac{9}{4\varepsilon }M\left(\phi \right)\nabla \mu \right|}_n\right\}+B{\left({\left|k\right|}^2\right)}^2{\widehat{\phi}}^n-S{\left|k\right|}^2{\widehat{\phi}}^n\right) $$

(14)