Deep Learning for the Modeling and Inverse Design of Radiative Heat Transfer

Deep learning is having a tremendous impact in many areas of computer science and engineering. Motivated by this success, deep neural networks are attracting increasing attention in many other disciplines, including the physical sciences. In this work, we show that artificial neural networks can be successfully used in the theoretical modeling and analysis of a variety of radiative-heat-transfer phenomena and devices. By using a set of custom-designed numerical methods able to efficiently generate the required training data sets, we demonstrate this approach in the context of three very different problems, namely (i) near-field radiative heat transfer between multilayer systems that form hyperbolic metamaterials, (ii) passive radiate cooling in photonic crystal slab structures, and (iii) thermal emission of subwavelength objects. Despite their fundamental differences in nature, in all three cases we show that simple neural-network architectures trained with data sets of moderate size can be used as fast and accurate surrogates for doing numerical simulations, as well as engines for solving inverse design and optimization in the context of radiative heat transfer. Overall, our work shows that deep learning and artificial neural networks provide a valuable and versatile toolkit for advancing the field of thermal radiation.

Figure 1

(a) A schematic representation of a neuron, the basic component of a neural network. The value of a neuron is determined by a linear transformation that weights the importance of various inputs, followed by a nonlinear activation function. Two typical nonlinear activation functions are also shown: sigmoid and ReLU. (b) A feed-forward neural network with neurons arranged into layers, with the output of one layer serving as the input to the next layer.

Figure 2

(a) A sketch of two identical multilayer systems separated by a vacuum gap of size $d_0$. The two reservoirs feature $N$ total layers alternating between a Drude metal (gray areas) with permittivity ${ϵ}_m$ and a dielectric (white areas) with permittivity ${ϵ}_d$. The last layer in both cases is made of metal and the thickness of layer $i$ is denoted by $d_i$. (b) The transmission of the evanescent waves as a function of the frequency $\omega$ and the magnitude of the parallel wave vector $k$ for the bulk system, i.e., two parallel plates made of the metal, and $d_0=10\mathrm{nm}$. (c) The same as in (b), but for the multilayer system with $N=160$ and $d_i=10\mathrm{nm}$ for all layers.

Figure 3

Results for the spectral heat-transfer coefficient $h_{\omega }$ between two multilayers with $N=4$ and a gap size $d_0=10\mathrm{nm}$. (a) A comparison between real $h_{\omega }$ spectra computed with fluctuational electrodynamics (solid lines) and the prediction of the NN (dashed lines). The layer thicknesses (in nanometers) are indicated in the legend. (b) A comparison of the NN approximation to the target spectrum (layer thicknesses in nanometers are indicated in the legend), following the inverse-design problem described in the text. (c) The result of the optimization problem in which the total heat-transfer coefficient is maximized. (d) The result of the optimization problem where $h_{\omega }$ is minimized in the frequency region indicated by the dashed vertical lines.

Figure 4

(a)–(d) The same as in Fig. 3 but now for $N=6$. (e)–(h) The same as in Fig. 3 but for $N=8$.

Figure 5

(a) Schematics of the silica mirror used as a passive radiative cooler in Ref. [69]. It consists of a $\mathrm{Si}\mathrm{O}$${}_2$ slab of thickness $d_{{\mathrm{Si}\mathrm{O}}_2}$ and a silver thin film of thickness $d_{\mathrm{Ag}}$. (b) A nanostructured version of the cooler shown in (a) but featuring a periodic array of circular holes of radius $R$ with a lattice parameter $a$ (square lattice). (c) The emissivity as a function of the wavelength for a silica photonic crystal (black solid line) and a silica mirror (red solid line) for $d_{{\mathrm{Si}\mathrm{O}}_2}=500\mu \mathrm{m}$ and $d_{\mathrm{Ag}}=120\mathrm{nm}$. For the photonic crystal, $a=100\mathrm{nm}$ and $R=50\mathrm{nm}$ ($f=0.785$). The cyan solid line corresponds to the AM1.5 solar spectrum $I_{\mathrm{AM1.5}}$ (see right vertical axis), the orange curve to the atmospheric emissivity or absorptivity spectrum ${\epsilon }_{\mathrm{atm}}$, and the gray dashed line to the black-body radiation curve $I_{\mathrm{BB}}$ (50 times enlarged in spectral irradiance) at 300 K.

Figure 6

(a) The emissivity of the photonic crystal cooling device as a function of the wavelength for several filling factors $f$ and a hole radius $R=50\mathrm{nm}$. The remaining parameters are $d_{{\mathrm{Si}\mathrm{O}}_2}=500\mu \mathrm{m}$ and $d_{\mathrm{Ag}}=120\mathrm{nm}$. (b) The corresponding cooling power density as a function of the device temperature under AM1.5 illumination. The ambient temperature of the atmosphere is taken to be $T_{\mathrm{amb}}=300\mathrm{K}$.

Figure 7

(a)–(c) A comparison between the emissivity spectra of silica photonic crystal devices computed with the RCWA method (solid lines) and the prediction of the NN (dashed lines) for different values of the silica-layer thickness and filling factor, as indicated in the legends. The hole radius is $R=30\mathrm{nm}$ and $d_{\mathrm{Ag}}=120\mathrm{nm}$.

Figure 8

(a) The cooling power density as a function of the silica-layer thickness and the filling factor of photonic crystal cooling devices as computed with the trained NN. In this case, the device temperature and the ambient temperature are assumed to be 300 K. The contribution of nonradiative processes is neglected. The black dot indicates the point for which the cooling power density reaches its maximum. (b) The corresponding equilibrium temperature. Note that the horizontal axes are inverted with respect to (a) to improve visibility.

Figure 9

(a) The emissivity of a silica cube of side $L$ as a function of the frequency. The dotted orange line corresponds to the Planck distribution $I_{\mathrm{BB}}(\omega ,T)$ at $T=300\mathrm{K}$ in arbitrary units. (b) The total power emitted by a $\mathrm{Si}\mathrm{O}$${}_2$ cube (see inset) as a function of its side $L$ at $T=300\mathrm{K}$. The power is normalized with the black-body result, $A\sigma T^4$, where $A=6L^2$ is the total area of the cube.

Figure 10

(a) A comparison between the real emissivity spectra of a $\mathrm{Si}\mathrm{O}$${}_2$ cube computed with TDDA (solid lines) and the prediction of the NN (dashed lines) for various values of the cube side $L$. (b) The corresponding comparison for the total power emitted by a $\mathrm{Si}\mathrm{O}$${}_2$ cube as a function of its side $L$ at $T=300\mathrm{K}$. The power is normalized with the black-body result.

Figure 11

(a) A comparison of the NN approximation for the emissivity of a $\mathrm{Si}\mathrm{O}$${}_2$ cube to a target spectrum for $L=10.250\mu \mathrm{m}$ following the inverse-design problem described in the text. (b),(c) The result of the optimization problem, where the emissivity of a $\mathrm{Si}\mathrm{O}$${}_2$ cube in the frequency range defined by the vertical dashed lines is maximized while the emissivity outside is minimized. The obtained value of the optimal cube side is indicated in the panels. (d) Learning curves showing the mean relative error (training and validation) as a function of the training set size for the optimal NN simulating the emissivity of a $\mathrm{Si}\mathrm{O}$${}_2$ cube.