Cosmology-informed neural networks to solve the background dynamics of the Universe

The field of machine learning has drawn increasing interest from various other fields due to the success of its methods at solving a plethora of different problems. An application of these has been to train artificial neural networks to solve differential equations without the need of a numerical solver. This particular application offers an alternative to conventional numerical methods, with advantages such as lower memory required to store solutions, parallelization, and, in some cases, a lower overall computational cost than its numerical counterparts. In this work, we train artificial neural networks to represent a bundle of solutions of the differential equations that govern the background dynamics of the Universe for four different models. The models we have chosen are $\mathrm{\Lambda }\mathrm{CDM}$, the Chevallier-Polarski-Linder parametric dark energy model, a quintessence model with an exponential potential, and the Hu-Sawicki $f(R)$ model. We use the solutions that the networks provide to perform statistical analyses to estimate the values of each model’s parameters with observational data; namely, estimates of the Hubble parameter from cosmic chronometers, type Ia supernovae data from the Pantheon compilation, and measurements from baryon acousstic oscillations. The results we obtain for all models match similar estimations done in the literature using numerical solvers. In addition, we estimate the error of the solutions that the trained networks provide by comparing them with the analytical solution when there is one, or to a high-precision numerical solution when there is not. Through these estimations we find that the error of the solutions is at most $\sim 1\%$ in the region of the parameter space that concerns the 95% confidence regions that we find using the data, for all models and all statistical analyses performed in this work. Some of these results are made possible by improvements to the method of solving differential equations with artificial neural networks conceived in this work.

Figure 1

Illustration of an example implementation of ANN-based bundle solutions to a differential system of a model $\mathcal{M}(\mathit{p})$ in a statistical analysis with a data set $\mathcal{D}$ to obtain samples from the posterior $\mathbb{P}(\mathit{p}|\mathcal{D})$.

Figure 2

Percentage error of the neural network–based bundle solution to $\mathrm{\Lambda }\mathrm{CDM}$ in its training range, through comparison to the analytical solution.

Figure 3

Percentage error of the neural network–based bundle solution to the CPL parametric dark energy model in its training range of $z$, a range of ${\omega }_1$ covering the range used in the statistical analysis, and four different values of ${\omega }_0$. The percentage error is calculated through comparison to the analytical solution.

Figure 4

Percentage error of the neural network–based bundle solutions of $H(z)/H_0^{\mathrm{\Lambda }}$ in the quintessence model, through comparison to numerical solutions. The range of the comparison goes through the training range of the parameters of the bundle.

Figure 5

Percentage error of the neural network–based bundle solution of $H(z)/H_0^{\mathrm{\Lambda }}$ in the Hu-Sawicki model, through comparison to numerical solutions. The range of the comparison goes through a section of the training range of the parameters of the bundle.

Figure 6

Results of the statistical analysis for the quintessence model with an exponential potential. The darker and brighter regions correspond to the 68% and 95% confidence regions, respectively. The plots on the diagonal show the posterior probability density for each of the free parameters of the model.

Figure 7

Results of the statistical analysis for the $f(R)$ Hu-Sawicki model. The darker and brighter regions correspond to the 68% and 95% confidence regions respectively. The plots on the diagonal show the posterior probability density for each of the free parameters of the model.

Figure 8

Top: $H(z)$ as a function of $z$ for all the models studied (using the parameters corresponding to the best fit obtained from the combination $\mathrm{CC}+\mathrm{SNIa}+\mathrm{BAO}$) together with CC data. Bottom: relative difference between the values of $H$ in the top panel for each model with respect to $\mathrm{\Lambda }\mathrm{CDM}$.

Figure 9

Top: apparent magnitude of a supernova, $m_b$, as a function of $z$ for all the models studied (using the parameters corresponding to the best fit obtained from the combination $\mathrm{CC}+\mathrm{SNIa}+\mathrm{BAO}$) together with SNIa data. The relation between $m_b$ and the Hubble parameter can be seen by looking at Eqs. (56), (58), and (59). Bottom: relative difference between the values of $m_b$ in the top panel for each model with respect to $\mathrm{\Lambda }\mathrm{CDM}$.

Figure 10

Example of a plot of the total loss per iteration of training. In blue are the values of the total loss at each iteration, and in red the average value of the total loss per 400 iterations. This particular example comes from training an ANN to solve $\mathrm{\Lambda }\mathrm{CDM}$ [Eq. (7)], defining the reparametrization in Eq. (a1) and the loss in Eq. (a3).