Approximating Rayleigh Scattering in Exoplanetary Atmospheres using Physics-informed Neural Networks (PINNs)

TL;DR

This study applies parameterized physics-informed neural networks (PINNs) to radiative transfer in exoplanetary atmospheres, focusing on Rayleigh scattering in an isothermal transit geometry. By embedding the stationary radiative transfer equation into the PINN loss and training separate models for absorption and scattering, the authors demonstrate that an absorption PINN can predict transmission spectra with RMSE on the order of a few percent and residuals around $\mathcal{O}(10^{-2})$, while a scattering PINN decomposes the solution into unscattered and scattered components with comparable accuracy. The work highlights both the potential and current limitations of PINNs for forward RT modeling, noting that speed gains over traditional numerical methods are not yet realized and that extensions to non-isothermal TP profiles, clouds/hazes, and multi-angle scattering are necessary for broader applicability. Overall, this approach offers a flexible, physics-guided pathway to efficient RT computations in diverse exoplanetary atmospheres and motivates further architectural and physical enhancements to achieve practical speedups.

Abstract

This research introduces an innovative application of physics-informed neural networks (PINNs) to tackle the intricate challenges of radiative transfer (RT) modeling in exoplanetary atmospheres, with a special focus on efficiently handling scattering phenomena. Traditional RT models often simplify scattering as absorption, leading to inaccuracies. Our approach utilizes PINNs, noted for their ability to incorporate the governing differential equations of RT directly into their loss function, thus offering a more precise yet potentially fast modeling technique. The core of our method involves the development of a parameterized PINN tailored for a modified RT equation, enhancing its adaptability to various atmospheric scenarios. We focus on RT in transiting exoplanet atmospheres using a simplified 1D isothermal model with pressure-dependent coefficients for absorption and Rayleigh scattering. In scenarios of pure absorption, the PINN demonstrates its effectiveness in predicting transmission spectra for diverse absorption profiles. For Rayleigh scattering, the network successfully computes the RT equation, addressing both direct and diffuse stellar light components. While our preliminary results with simplified models are promising, indicating the potential of PINNs in improving RT calculations, we acknowledge the errors stemming from our approximations as well as the challenges in applying this technique to more complex atmospheric conditions. Specifically, extending our approach to atmospheres with intricate temperature-pressure profiles and varying scattering properties, such as those introduced by clouds and hazes, remains a significant area for future development.

Figures (9)

• Figure 1: Left: The transit geometry. The x (and y) axis was scaled by a factor of $l_x$ (and $l_y$), such that both coordinates are normalized. Right: Example differential optical depth $\frac{\mathrm{d} \tau}{\mathrm{d} x} = l_x \alpha(x,y)$, where $\alpha \sim P$.
• Figure 2: Testing the $\alpha_a(P)$-profile generation algorithm by comparing the resulting distribution of logarithmic gradients between every layer with the same distribution for all opacity sources of the example spectrum shown in Fig. \ref{['fig:exampleSpec_absPINN']}. The portion of gradients of the example spectrum that lay outside of the shown interval accumulate to less than 0.1% for $\frac{\mathrm{d}\ln(\alpha_a)}{\mathrm{d}\ln(P)} < -0.5$ and to about 1.1% for $\frac{\mathrm{d}\ln(\alpha_a)}{\mathrm{d}\ln(P)} > 3.5$.
• Figure 3: The architecture of the scattering PINN, where the numbers in the (fully-connected) layers indicate their width. The 4th layer connects both to an output $u_a$ and to two subsequent layers that have an output $u_s$. Because $u_a$ models the part of the solution that fulfills the RTE without scattering, a filter is applied, dependent on $\phi$, such that $u_a = 0$ where the light cannot get without changing directions. The combined output $u_a + u_s$ should fulfill the complete RTE, such that $u_s$ only has to model the scattered light component of the solution. Since the diffuse light only becomes relevant for large enough $\Delta_*$, $u_s$ is multiplied with the solid angle subtended by the star.
• Figure 4: Violin plot of the 95 root mean squared errors of each test spectrum for each trained absorption PINN. The maximum RMSE of the 4x8 PINN lies outside the shown range at 7.7%.
• Figure 5: Example for the evolution of the different loss components, for the 3rd 4x64 PINN. The total loss $\mathcal{L}$ is calculated from the boundary ($\mathcal{L}_\mathcal{B}$) and residual loss ($\mathcal{L}_\mathcal{F}$) as $\log_{10}(\mathcal{L}) = \log_{10}(\mathcal{L}_\mathcal{B} + 0.5 \mathcal{L}_\mathcal{F})$, as defined in Eq. \ref{['eq:lossComponents']} - \ref{['eq:boundaryLoss']}.