Information field dynamics for simulation scheme construction

Information field dynamics (IFD) is introduced here as a framework to derive numerical schemes for the simulation of physical and other fields without assuming a particular subgrid structure as many schemes do. IFD constructs an ensemble of nonparametric subgrid field configurations from the combination of the data in computer memory, representing constraints on possible field configurations, and prior assumptions on the subgrid field statistics. Each of these field configurations can formally be evolved to a later moment since any differential operator of the dynamics can act on fields living in continuous space. However, these virtually evolved fields need again a representation by data in computer memory. The maximum entropy principle of information theory guides the construction of updated data sets via entropic matching, optimally representing these field configurations at the later time. The field dynamics thereby become represented by a finite set of evolution equations for the data that can be solved numerically. The subgrid dynamics is thereby treated within auxiliary analytic considerations. The resulting scheme acts solely on the data space. It should provide a more accurate description of the physical field dynamics than simulation schemes constructed ad hoc, due to the more rigorous accounting of subgrid physics and the space discretization process. Assimilation of measurement data into an IFD simulation is conceptually straightforward since measurement and simulation data can just be merged. The IFD approach is illustrated using the example of a coarsely discretized representation of a thermally excited classical Klein-Gordon field. This should pave the way towards the construction of schemes for more complex systems like turbulent hydrodynamics.

Figure 1

A realization of a thermally exited KG field ${\phi }_x$ (a) and its momentum distribution ${\pi }_x$ (b) is shown for $\beta =1$ and $\mu =1$ at $t=0$ with a resolution of 2048 pixels with black lines passing through the diamond symbols. The low-resolution data with $\mathcal{N}=64$ data points describing the same fields are shown with yellow diamonds. The field configuration at $t=0.1$ is also shown in panel (a) with a thin brown (gray) line. The KG field ${\phi }_x$ shows a correlated structure due to the suppression of small scale power by the gradient term in the Hamiltonian, whereas its momentum field ${\pi }_x$ is just white noise. The loss of small-scale structure information in the low-resolution sampling is especially apparent for the momentum data.

Figure 2

(a) Fourier–data-space dispersion relations ${\tilde{\omega }}_k$ of numerical schemes for the KG field simulation for the parameters $\mathcal{N}=64$ and $\mu =1$. The IFD scheme data mode frequencies ${\check{\omega }}_{k,t}$ are shown at initial time $t=0$ as given by (69) (top, blue dots), an instance later at $t={10}^{-4}$ (top, blue solid line with kinks), and at time $t=\pi /2$ (strongly oscillating blue dotted line). At $t=\pi ,$ the IFD scheme dispersion relation looks similar to the initial one. The spectral scheme frequencies ${\tilde{\omega }}_k^{\mathrm{spec}}$ as given by (71) (middle, black squares) follow the continuous-space field dispersion (thin, smooth, and black line). Finally, the finite-difference scheme ${\tilde{\omega }}_k^{\mathrm{diff}}$ as given by (70) has the lowest frequencies (bottom, brick red triangles). (b) Data-space representation of the numerical scheme operator ${\tilde{L}}_{ij}$ as a function of the pixel number difference $i-j$ for small differences. The curves are given by the discrete Fourier transformations of ${\tilde{\omega }}_k^2$ for the IFD scheme at $t=0$ (most extreme, blue dots and line) as well as for $t=\pi /2$ (smaller light blue dots and blue dotted line close to intermediate black line), the spectral scheme (intermediate values, black squares and line), and for the finite-difference scheme (most moderate values, brick red triangles and line). It should be noted that the IFD operator at $t=\pi /2$ also contains some power around positions $i-j=\pm \mathcal{N}/2=\pm 32$ (not shown in this figure) as a consequence of the heavy oscillations of ${\check{\omega }}_{k,t}$ at this time that are visible in panel (a).

Figure 3

(a) Evolved field (thin, black line) and data at $t=10$ of the field also shown in Fig. 1 ($\beta =1,\mu =1,\mathcal{N}=64$). The exact data ${\tilde{d}}_t=R{\phi }_t$ are shown as yellow diamonds. The IFD data according to (66) and (67) (blue dots) follows the exact data closely. The data of the spectral scheme (black squares) is very close to the IFD data. The data of the difference scheme (brick red triangles) exhibit the poorest match to the correct data of the evolved field. The root-mean-square errors of the field data values ${\sigma }_d^{\left(\phi \right)}=\sqrt{{\sum }_{i=0}^{\mathcal{N}-1}{({\tilde{d}}^{\left(\phi \right)}-R\phi )}_i^2/\mathcal{N}}$ of the three schemes are $0.003$, $0.004$, and $0.020$ for the IFD, spectral, and difference schemes, respectively. (b) Temporal evolution of the data error ${\sigma }_d^{\left(\phi \right)}\left(t\right)$ for the IFD (bottom solid blue line), spectral (dashed black line slightly above the former), and finite-difference (top brick red line) schemes. The dip in the IFD and spectral scheme error at $t=\pi$ is due to the nearly perfect alignment of the mode phases at this particular time.