Field dynamics inference via spectral density estimation

Stochastic differential equations are of utmost importance in various scientific and industrial areas. They are the natural description of dynamical processes whose precise equations of motion are either not known or too expensive to solve, e.g., when modeling Brownian motion. In some cases, the equations governing the dynamics of a physical system on macroscopic scales occur to be unknown since they typically cannot be deduced from general principles. In this work, we describe how the underlying laws of a stochastic process can be approximated by the spectral density of the corresponding process. Furthermore, we show how the density can be inferred from possibly very noisy and incomplete measurements of the dynamical field. Generally, inverse problems like these can be tackled with the help of Information Field Theory. For now, we restrict to linear and autonomous processes. To demonstrate its applicability, we employ our reconstruction algorithm on a time-series and spatiotemporal processes.

Figure 1

Data $d$ (a), signal $\varphi$ (b), and reconstruction $m_{\varphi }$ (c) using the MAP estimate of the spectrum. The gray area denotes the one $\sigma$ uncertainty of the reconstruction.

Figure 2

Reconstruction of $P_{\varphi }$, given $\varphi$ (Fig. 1, middle panel), on a log-log-scale. The solid line is the theoretical spectrum corresponding to Eq. (36) and the black dots display the spectrum of the actual sample $\varphi$. The dashed line is the maximum a posteriori estimate obtained by maximizing Eq. (24) and the dotted line is the MAP estimate of $\tau$ alone. The gray area denotes the one $\sigma$ uncertainty of the reconstruction as described in Sec. 3e.

Figure 3

Reconstruction of $P_{\varphi }$ given noisy data $d$ (Fig. 1, top panel). We note that due to the Gaussian approximation the uncertainty estimates are significantly underestimated, in regions of low power.

Figure 4

Signal $\varphi$ [(a), drawn form the process corresponding to Eq. (37)], resulting noisy measurement data $d$ (c), reconstruction $m_{\varphi }$ (b), and uncertainty map $\sqrt{\widehat{D}}$ (d) of the reconstruction.

Figure 5

For the field and data shown in Fig. 4, logarithmic spectrum $ln\left(P_{\varphi }\right)$ (a), projected data $ln\left({\left|d\right|}^2\right)$ (c), reconstruction $\tau +tan\left(\delta \right)$ (b), and uncertainty estimate $\sqrt{\widehat{O}}$ (d) as defined in Sec. 3e.

Figure 6

Slices through the logarithmic spectrum, the data, and the reconstruction, shown in Fig. 5. The left panels show the spectrum as a function of $k$ for fixed $\omega =0$ (a) and $\omega =-30$ (c). The right panels show the spectrum as a function of $\omega$ for fixed $k=-20$ (b) and $k=70$ (d).

Figure 7

Slices through the full field, the data, and the reconstruction, shown in Fig. 4. The left panels show the spatial structure for $t=-0.27$ (a) and $t=0.38$ (c). The right panels show the temporal evolution at $x=-0.15$ (b) and at $x=0.17$ (d).

Figure 8

Signal $\varphi$ (a), noisy measurement data $d$ (c), reconstruction $m_{\varphi }$ (b), and uncertainty map $\sqrt{\widehat{D}}$ (d), for the highly structured spectrum given by Eq. (38).

Figure 9

For the field and data shown in Fig. 8, logarithmic spectrum $ln\left(P_{\varphi }\right)$ (a), projected data $ln\left({\left|d\right|}^2\right)$ (c), reconstruction $\tau +tan\left(\delta \right)$ (b), and uncertainty estimate $\sqrt{\widehat{O}}$ (d).