Inference of neutrino flavor evolution through data assimilation and neural differential equations

The evolution of neutrino flavor in dense environments such as core-collapse supernovae and binary compact object mergers constitutes an important and unsolved problem. Its solution has potential implications for the dynamics and heavy-element nucleosynthesis in these environments. In this paper, we build upon recent work to explore inference-based techniques for the estimation of model parameters and neutrino flavor evolution histories. We combine data assimilation, ordinary differential equation solvers, and neural networks to craft an inference approach tailored for nonlinear dynamical systems. Using this architecture, and a simple two-neutrino-beam, two-flavor model, we compare the performances of nine different optimization algorithms and expand upon previous assessments of the efficacy of inference for tackling problems in flavor evolution. We find that employing this new architecture, together with evolutionary optimization algorithms, accurately captures flavor histories in the small-scale model and allows us to quickly explore both model parameters and initial flavor content. In future work we plan to extend these inference techniques to large numbers of neutrinos.

Figure 1

Data assimilation as a neural network architecture. The forward problem is represented by a set of first-order ordinary differential equations, $\mathit{F}(\mathit{u},r,\mathit{\theta })$, that depend on the unknown model parameters $\mathit{\theta }$. $\mathrm{Cost}(\mathit{\theta })$ is the objective function whose minimum denotes the optimal solution of the inverse problem. Blue arrows denote the forward process of prediction, and red arrows denote the backward pass for error correction.

Figure 2

The solution to the forward problem, in terms of the $z$ components of the polarization vectors, ${\mathbf{P}}_{\mathbf{z}}$, of the two neutrino beams, as functions of the affine parameter $\mathbf{r}$. The parameters used in the forward integration are shown in Table 1. One neutrino beam (orange) is initially electron flavor, and the other (blue) is initially x flavor. In this toy problem, we assume the detector to be at location $r=5$. For solving the inverse problem, the only information available to the procedure is the polarization measured at this location.

Figure 3

Cost function dependence on the free parameter $V_0$, where the simulated “detector” data was obtained through forward integration using the parameter values of Table 1. The global minimum is obtained at $V_0={\tilde{V}}_0$ (see Table 1), but there are many local minima present. Thus, even with only a single unknown parameter, this problem presents a very difficult challenge for the optimization.

Figure 4

Gradient (orange) and Hessian (green) of cost function with respect to the free parameter $\theta =V_0$. Ideally, one expects that these derivatives will be large at locations far from the global optimum, and that they will gradually go to zero as the minimum is approached. In our setup there is a sudden transition from large oscillations to tiny ones near $V_0={\tilde{V}}_0$. For small $V_0$, gradient based methods will have very large parameter updates in either direction, and for large $V_0$ the updates will be very small.

Figure 5

The cost function at the final inferred values of $V_0$ (i.e., at the end of the optimization procedure) as function of the initial guess $V_0^{\text{initial}}$, for each of the nine optimization algorithm. Ideally this value should be 0, and in practice, the smaller it is the better the optimization. For visual clarity, the nine algorithms are plotted in two groups 5 and 5. In this set of experiments, the optimization procedure is instructed to regard $V_0$ as a constant unknown parameter.

Figure 6

The final inferred value of $V_0$ as function of the initial guess $V_0^{\text{initial}}$, for each optimization algorithms. For a converged optimization this value should be ${\tilde{V}}_0=50$, independent of the initial guess (see Table 1). These results are for the same set of experiments as those in Fig. 5.

Figure 7

The minimum, maximum and average values of the cost function at the final values of $\mathit{\theta }$ (i.e., at the end of the optimization procedure), for each optimization algorithm. Ideally, all these three values should be 0. The spread shows how dependent the optimization algorithms are on the initial guess of the unknown parameters. In this set of experiments, the optimization procedure treats $V_0$ as a generic function, encoded in terms of a two-layer feed-forward neural network, that depends on unknown parameters $\mathit{\theta }$.

Figure 8

Optimal matter potential coupling $V_0$ found by each method as function of the affine parameter $r$. Some algorithms find the coupling to be a constant close to the value in Table 1, while others show sharp matter profile changes. This shows that various matter profiles, despite being quite different, lead to very similar values of the polarization $z$-components detected at $r=5$. This is for the same set of experiments described in the caption of Fig. 7.

Figure 9

Dependence of the cost function, Cost $({\theta }_{1_P},{\theta }_{1_A})$, on the initial orientation in flavor space of the polarization vector of the first neutrino. There is a relatively strong dependence on the polar angle, ${\theta }_{1_P}$, and weak dependence on the azimuthal angle, ${\theta }_{1_A}$. This is an indication of degeneracy in parameter space. Here, the matter potential was taken to be a known constant ${\tilde{V}}_0$ (see Table 1), and only the initial angles ${\theta }_{1_P}$ and ${\theta }_{1_A}$ were varied. The simulated “detector” data was generated by using $P_{1,z}=-1$ as the starting point of the forward integration.

Figure 10

The minimum, maximum and average values of the cost function at the final values of $\mathit{\theta }$ (i.e., at the end of the optimization procedure), for each optimization algorithm. Ideally, all these three values should be 0. For each algorithm, the spread of cost function values shows how dependent the optimization algorithms are on the initial guess of the unknown parameters. These results are for the set of experiments described in the caption of Table 2, where the unknown parameters to be optimized are $\mathit{\theta }=\{{\theta }_{1_P},{\theta }_{1_A}\}$.

Figure 11

The minimum, maximum and average values of the cost function at the final values of $\mathit{\theta }$ (i.e., at the end of the optimization procedure), for each optimization algorithm. Ideally, all these three values should be 0. For each algorithm, the spread of cost function values shows how dependent the optimization algorithms are on the initial guess of the unknown parameters. These results are for the set of experiments where the matter coupling parameter is taken to be a known constant, and the unknown parameters to be optimized are the polar and azimuthal angles of both the initial neutrino polarization vectors, i.e., $\mathit{\theta }=\{{\theta }_{1_P},{\theta }_{1_A},{\theta }_{2_P},{\theta }_{2_A}\}$.

Figure 12

The final inferred value of $V_0$ (green) with the APS optimization method, and its corresponding cost value (red), as function of the initial guess $V_0^{\text{initial}}$. In this experiment, the optimization procedure was given a model with the neutrino-neutrino coupling ${\mu }_0=0$, even though the simulated “detector” data was generated using a model with ${\mu }_0=10$. The lack of a global convergence and the relatively high values of the cost function are indications that the model with ${\mu }_0=0$ does not properly reproduce the detector measurements.