physics.comp-phOct 8, 2026
SaveGhost tasking for parametrized Gaussian Processes solving linear differential equations
Abstract
Physics-informed machine learning has gained significant attention in recent years. In regimes of limited data, parametrized Gaussian processes have become popular. Existing approaches, however, often face limitations, such as requiring parametrizable (also called controllable) systems or a large number of output tasks. In this work, we introduce a systematic procedure we call "ghost tasking", using auxiliary tasks to circumvent these limitations. We prove that such ghost tasks can render any non-parametrizable system effectively parametrizable, enabling algorithmic construction of parametrized Gaussian Processes while keeping the number of required tasks (i.e. output dimensions) and latent functions low. We find that ghost tasking performs especially well in an inverse problem setting, even with very few available data. We show the usage and power of ghost tasking in three experiments, providing systematic comparisons to the only other currently available method applicable to all experiments. We provide necessary syntax and explications for two computer algebra programs that compute parametrizations for systems with polynomial or rational coefficients. Our theoretical results extend to systems with meromorphic functions.
Figures & tables
Figure 2: Overview over parametrizability in operator space and function space . Every combination can be read as "Systems represented by modules in operator space that are [i.e. torsion-free] can parametrize [i.e. all] solutions over the function space if [i.e. is injective]"
Figure 3: Overview over the idea behind ghost tasking and its connection to physics-informed Gaussian Processes (PIGP) as in [ 43 ] . A non parametrizable system can be rendered parametrizable by adding extra tasks, and subsequently conditioning them away again. The differences between ghost tasking and PIGPs lie in (a) how many tasks we add and (b) how they are coupled to the system ( versus the identity matrix ).
Figure 4: Inferred solutions of the forward problem of experiment 1 for the GT method (left), GT+ with extra collocation points (middle), and the PIGP method (right). Each subplot shows the solution of one task, the last three are the derivative of the angles in the first three tasks, necessary to constrain a second order initial condition problem. The true solution is marked in red, the training data with black stars, and the inferred solutions in blue with a lightblue uncertainty band. The RMSE on the test points is written in the corner of each task.
Figure 5: Results of the systematic comparison in experiment 1 of GT (green), GT+ (orange), and PIGP (blue) using the mean absolute error of the learned physical parameters (maximum a posteriori, MAP value) (first row), (middle row), as well as their ratio (last row) relative to their true values as metric. Note that is not an independent parameter, but does dictate much of the dynamics of the system and can as such be most precisely estimated by GT+. The experiments have been conducted for different numbers of training data for the tasks, as well as different noise levels , shown in different subplots. We show the distinct runs as scatterplots, as well as their box-plots for each combination , n).
Figure 6: Re-scaled log posteriors in the -plane marginalized over the calibration parameters and for all combinations of and . The reference Bayesian estimate for the known solution (upper left plot) is compared to GT (upper right), GT+ (lower right), and PIGP (lower left). The elongated structures show a strong correlation between and , and we see multimodality in the reference and GT posteriors. We do not show colorbars because the likelihood scales vary substantially between panels; the comparison is qualitative and focuses on the relative structure of each likelihood rather than absolute values compared across panels.
Figure 7: Inferred solution (uneven rows) and absolute error (even rows) of the forward problem given training data for the ghost task/input tasks and the initial conditions. The torsion-free GT version is depicted in the first two rows, the projective GT version in the middle two rows, and PIGP in the lowest two rows. The left column shows the results for , the middle column for and on the right we show the ghost task resp. input tasks for the inhomogeneity necessary. Conditioned points are marked with blue. The RMSE is computed within the region of interest, marked by a red box.
Figure 8: Relative error as performance metric of the inverse problem of experiment 2, analogously to Fig. 5 of experiment 1. The results for the torsion-free GT system are shown in green, the results for the projective GT version are shown in orange and the results for PIGP are shown in blue.
Figure 9: Quiver plot of the magnetic flux density inferred with GT in the forward problem of experiment 3 given Dirichlet boundary conditions for and training points for and the ghost task . Direction of is indicated with arrows, the strength is represented with the length of the arrows as well as their color.
Figure 10: Visualization of the conditioned tasks in the forward problem of Experiment 3. The left column shows two dimensional slices though the plane at of the analytic ground truth, the middle column shows the ghost tasking method, and the right column shows PIGP. The rows depict the (x,y,z) components of the vector potential , the component of the current density , and the ghost task for GT, and , respectively for PIGP. Training data is marked with blue points and the RMSE over the full volume is written in the corner of each task.
Figure 11: Visualization of the solution of the forward problem of Experiment 3 for the magnetic flux density components. The left column shows two dimensional slices though the plane at of the analytic ground truth, the middle column shows the ghost tasking method, and the right column shows PIGP. The rows depict the (x,y,z) components of the inferred quantity of interest, the magnetic flux density . No training data on is used. The RMSE over the full volume is written in the corner of each task.
Figure 12: Relative error of the MAP estimate of the parameter as performance metric of the inverse problem of experiment 3, analogously to Fig. 5 and 8 . We see a clear advantage of the GT method (green) as compared to PIGP (blue) in all experiments.
Figure 13: Re-scaled components of the log marginal likelihood (Eq. 2 ) for GT (marked with x) and PIGP (marked with circles) as a function of , conditioned on the respective MAP estimates of the calibration parameters and for the training data set and all values evaluated on 50 points . The first row shows the complexity term, the second row shows the data-fit term, and the last row shows the total log marginal likelihood. Curves are not normalized due to re-scaling; the relevant information is conveyed by the position and width of the peaks.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
| measured | known | |||||
|---|---|---|---|---|---|---|
| GT | PIGP | GT | PIGP | |||
| n | standard | + | standard | |||
| 3 | 0.001 | / / | / / | / / | / / | / / |
| 3 | 0.01 | / / | / / | / / | / / | / / |
| 3 | 0.1 | / / | / / | / / | / / | / / |
| 3 | 0.2 | / / | / / | / / | / / | / / |
Table 1: Ablation study experiment 1. Median absolute error of the physical parameters (each cell: / / ) relative to their true values across 10 runs. We compare the variants "standard", where every task has the same amount of training data , and "+", where the ghost task is constrained on points and the standard deviation of the likelihood in these tasks is fixed to be . We compare two settings "measured ", where is provided with measurements, and "known ", where is provided with 25 noise-free input points for all settings. For PIGP, we cannot provide a "+" variant for "measured " per definition. Overall performance is calculated using the geometric mean of the above results and shown in the last row. We see similar performances between GT and PIGP, while "GT+" outperforms both drastically especially for in the measured case. For known , performances are similar.
Figure 14: Re-scaled log posteriors of analogously to Fig. 6 assuming is known. We recover similar multimodal structures as we did in the original experiment, but this time for both methods.
| GT (torsion-free) | GT (projective) | PIGP | |||||
| n | standard | + | standard | + | standard | + | |
| 3 | 0.001 | 0.000663676 | 0.000510805 | 0.000355886 | |||
| 3 | 0.01 | 0.00308351 | 0.0103383 | 0.00339176 | |||
| 3 | 0.1 | 0.0360237 | 0.0333033 | 0.044 | |||
| 3 | 0.2 | 0.0953646 | 0.168597 | 0.109904 | |||
| 5 | 0.001 | 0.000189513 | 0.000141994 | 0.000185464 | |||
Table 2: Ablation study experiment 2. Median absolute error of the physical parameter relative to its true value , analogously to 1 . For the "+" variant, the ghost task resp. the inhomogeneities (for GT resp. PIGP) are constrained on points and the standard deviation of the likelihood in these tasks is fixed to be . Contrary to experiment 1 ( 1 ), we do not see significant differences between variants in the results.
| GT | PIGP | ||||
| n | standard | + | standard | + | |
| 3 | 0.001 | 0.0809063 | 0.342445 | ||
| 3 | 0.01 | 0.0731214 | 0.345774 | ||
| 3 | 0.1 | 0.0539069 | 0.386825 | ||
| 3 | 0.2 | 0.0469981 | 0.434896 | ||
| 5 | 0.001 | 0.000884615 | 0.32439 | ||
Table 3: Ablation study experiment 3. Median absolute error of the physical parameter relative to its true value , analogously to 1 . For the "+" variant, the ghost task resp. the inhomogeneities (for GT resp. PIGP) are constrained on points and the standard deviation of the likelihood in these tasks is fixed to be . GT outperforms PIGP in every setting. Contrary to experiment 1 ( 1 ), we do not see significant differences between variants in the results.