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Ghost tasking for parametrized Gaussian Processes solving linear differential equations

Johanna Moser, Christopher Albert, Sascha Ranftl

arXiv:2610.12009v1physics.comp-phcs.LG

TL;DR

Existing parametrized Gaussian-process methods can require parametrizable systems or many output tasks. Ghost tasking adds auxiliary tasks to construct parametrizations for otherwise non-parametrizable systems, and the experiments report more informative inverse-problem fits with fewer latent functions. Its torsion-free variant carries a theoretical caveat: full solution-space parametrizability is not guaranteed.

  • Problem

    Existing parametrized-GP methods can require parametrizable systems or many output tasks, while PIGPs can lack robustness in low-data regimes.

    Source: §2.2 · p. 8

  • Method

    Ghost tasking adds at most two auxiliary tasks to lift a system into a parametrizable form, then conditions those tasks to zero.

    Source: §6 · p. 36

  • Results

    Across experiments, fewer latent functions more strongly constrain the data fit; in Experiment 3, GT+B uses one latent function across five tasks, versus PIGP's three across six tasks.

    Source: §5.1 · p. 34

  • Takeaways & Limitations

    Ghost tasking provides a way to extend parametrization to systems that are not parametrizable on their own while retaining a minimal latent function space.

    Source: §6 · p. 36

    The torsion-free variant does not guarantee parametrizability of the full solution space, though the authors did not find this limitation practically relevant in their experiments.

    Source: §5.2 · p. 35

Abstract

from arXiv · show

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.

Highlights

The paper presents ghost tasking as a way to balance accuracy and applicability in physics-informed Gaussian processes, connecting parametrized-GP methods to algebraic parametrization. It reports theoretical validity for full-row-rank systems with meromorphic coefficients and improved low-data inverse-problem accuracy.

Contribution

  • The work connects physics-informed Gaussian-process methods to Lange-Hegermann (2018).Source: Highlights · p. 2

Theory

  • Ghost tasking is proved valid for full-row-rank systems with meromorphic coefficients.Source: Highlights · p. 2

Results

  • Ghost tasking improves accuracy over Raissi (2017) in low-data inverse problems.The paper also reports systematic comparisons with that method.Source: Highlights · p. 2

Implementation

  • The paper includes tutorials for using OreModules and Macaulay2 to compute parametrizations.Source: Highlights · p. 2

1. Introduction

The introduction situates the work in hard-constrained physics-informed Gaussian processes and identifies limits in existing parametrization approaches. It introduces ghost tasking as a higher-dimensional construction intended to extend applicability while retaining performance in forward and inverse problems.

1. Introduction

  • Hard constraints in Gaussian processes enforce linear differential equations exactly through covariance structure while retaining uncertainty quantification.Source: §1 · p. 3

1.1. Related Work

  • Parametrized-GP approaches enforce linear differential equations through operator-transformed kernels, while Mercer-type GPs require known fundamental solutions.Other approaches include boundary-condition kernels [16] and soft constraints for nonlinear PDEs [24].Source: §1.1 · p. 4

1.2. Contribution

  • Existing methods can require parametrizability or constant coefficients, while Raissi et al. can lack robustness in low-data regimes.Source: §1.2 · p. 4
  • Ghost tasking lifts a non-parametrizable system with auxiliary output dimensions, parametrizes the lifted system, then conditions ghost tasks to zero.The paper proves validity for full-row-rank systems with polynomial, rational, or meromorphic coefficients, and reports improved inverse-problem accuracy, especially with little data.Source: §1.2 · pp. 4–5
  • The work identifies Raissi et al. (2017) and Lange-Hegermann (2018) [10] as points on one methodological spectrum and presents ghost tasking as a balance of accuracy and applicability.It reports three experiments, compatibility with PCGP, and tutorials for OreModules and Macaulay2.Source: §1.2 · p. 5

2. Background

The background introduces Gaussian-process tools and algebraic conditions used to construct parametrizations of differential-equation solutions. It contrasts existing parametrized-GP methods and explains when their parametrizations cover all solutions in a chosen function space.

2.1. Gaussian Processes

  • Multitask Gaussian processes couple outputs through linear transformations, allowing differential-equation constraints to be encoded in the kernel.A GP is specified by its mean and kernel, and conditioning on observations yields a GP at new inputs.Source: §2.1 · p. 6

2.3. A short introduction to some algebraic concepts

  • The operator rings include Weyl algebras for polynomial coefficients and rings for rational or meromorphic coefficients; the stated results cover these and certain ODE rings with holomorphic coefficients.Source: §2.2 · p. 7

2.2. Parametrized Gaussian Processes

  • PIGPs model unknown functions and inhomogeneities using 2q tasks, whereas null-space-based parametrization typically uses fewer tasks and latent functions.The reduced latent space can strengthen task coupling and help inverse-problem performance, but PIGPs can be less robust in low-data regimes.Source: §2.2 · p. 8
  • The null-space-based method transforms a base GP with Q, whose columns lie in the null space of the system matrix, to generate solutions.Its parametrization covers all solutions only when the system is parametrizable; otherwise it may generate only some solutions or, often, only the trivial solution.Source: §2.2 · p. 8

2.3. A short introduction to some algebraic concepts

  • For full-row-rank system matrices, module properties such as torsion-freeness and projectivity can be tested using extension modules or a right inverse.The paper relates torsion-freeness to vanishing of an extension module and projectivity to vanishing of all listed extension modules or existence of a right inverse.Source: §2.3 · p. 10

2.4. Algebraic parametrization

  • A torsion-free system admits a nontrivial Q with RQ=0, but all solutions are parametrized only if the function space is injective; projectivity also suffices.The paper notes that standard C∞ spaces are not injective for Weyl algebras or general meromorphic coefficients.Source: §2.4 · p. 10, §2.3 · p. 10
  • If the system has torsion, parametrization may cover only some solutions; for a torsion system, Q is trivial and yields only the zero solution.Source: §2.4 · p. 11
  • For torsion-free systems, minimal parametrizations have p−q columns, equal to the module rank.Source: §2.4 · p. 11

3. Ghost tasking

Ghost tasking turns otherwise non-parametrizable differential systems into parametrizable embedded systems using a small number of auxiliary tasks, then recovers the original solution space by conditioning those tasks to zero. The section relates this construction to PIGP and NSB, proves task bounds, and discusses practical refinements.

3.1. Bridging PIGP and NSB

  • PIGP is interpretable as a parametrization formed by applying a matrix to the base kernel, and its construction guarantees projectivity.Its per-unknown inhomogeneity construction uses 2q tasks, whereas Eq. 7 can use fewer for correlated inhomogeneities or q in the homogeneous case.Source: §3.1 · p. 12

3.2. Foundations of ghost tasking

  • Ghost tasking embeds the system, parametrizes the embedded null space, and conditions ghost tasks to zero; this recovers the original system pointwise.The construction makes the embedded system parametrizable before the auxiliary tasks are constrained away.Source: §3.2 · pp. 13, 15
  • The construction depends only on the residue class of Λ, so replacing Λ by Λ + RX leaves the result unchanged.Source: §3.2 · p. 14
  • For non-parametrizable systems, at most two ghost tasks yield a stably free, hence parametrizable, embedded system whose constrained solutions coincide with those of the original system.For ordinary differential-operator rings, one ghost task suffices.Source: §3.2 · pp. 14–15

3.3. Properties of ghost tasking

  • In practice, ghost tasks are conditioned to zero at finitely many points, and Λ can be found by computation or by testing low-order candidates.A small to moderate number of conditioning points is reported to suffice empirically.Source: §3.3 · pp. 15–16
  • A weaker variant requires only torsion-freeness rather than stable freeness, and it too can be achieved with at most two columns in Λ.The section notes this weaker requirement may be desirable for efficiency.Source: §3.3 · p. 15
  • The task bound also bounds latent functions: the number ranges from p−q to p, with one or two ghost tasks sufficing for homogeneous equations.The section gives n = (p + m) − q latent functions for the embedded system.Source: §3.3 · p. 16

4. Experiments

The experiments compare minimally tasked ghost tasking (GT) with PIGP on forward and inverse problems across a tripendulum ODE and two PDEs, examining additional zero-ghost-task collocation points and projective versus torsion-free parametrizations. In anisotropic magnetostatics, they also evaluate using auxiliary tasks to infer magnetic flux density from latent vector potential, with both methods performing well on the forward problem and GT performing significantly better on the inverse problem.

4.1. Experiment 1: Tripendulum

  • GT matches PIGP in low-data inference, improves g/ℓ performance by about one order of magnitude, and GT+ improves g/ℓ results by up to 5 orders of magnitude.GT+ uses additional ghost-task collocation points; the authors attribute its gains to suppressing oscillations absorbed as ghost-task residue.Source: §4.1 · p. 20
  • GT+ recovers the multimodal posterior structure seen in the reference Bayesian estimate, whereas PIGP is unimodal and can yield biased posteriors.GT also exhibits multimodality, but with a slightly shifted frequency; PIGP recovers the structure when the forcing term is nearly known exactly.Source: §4.1 · pp. 21–22, Figure 6 · p. 23

4.2. Experiment 2: Coupled transport

  • Both GT parametrization variants and PIGP achieve forward-problem RMSE below 0.006 on every task, with projective GT slightly outperforming the others while using less training data.Projective GT took about 25 seconds for inference, versus about 3 seconds for torsion-free GT and PIGP.Source: §4.2 · p. 25, Figure 7 · p. 26
  • In the inverse problem, projective and torsion-free GT recover PIGP’s performance while using less training data, but projective GT takes approximately 10 times longer to converge.The authors report comparable performance across settings, consistent with the system’s loose coupling.Source: §4.2 · p. 25, Figure 8 · p. 27

4.3. Experiment 3: Anisotropic Magnetostatics

  • For anisotropic magnetostatics, the system is not parametrizable; computing its projective parametrization was infeasible on a laptop, so the experiment uses a torsion-free construction.The passage states that the torsion-free system is sufficient to vastly outperform PIGP, but provides no result values in this part.Source: §4.3 · p. 27
  • Because the vector potential A is latent and unobservable, the experiment infers magnetic flux density B by taking the curl of A.The inverse problem does not use A because it is not measurable.Source: §4.3 · p. 28
  • In the inverse problem, GT performs significantly better than PIGP, with a sharper data-fit peak near the correct ν0 ≈1 offsetting a bias that PIGP’s flatter data-fit term cannot overcome.The complexity penalties are similarly peaked for both methods and favor small ν0; the authors suggest the difference may relate to parametrization-matrix column rank and coupling strength.Source: §4.3 · p. 29, Figure 12 · p. 32, Figure 13 · p. 33
  • RMSEs between 0.002 and 0.01 in the inferred magnetic flux density components show good forward-problem performance across both methods.The compared solutions infer B without training data on B.Source: §4.3 · p. 29, Figure 11 · p. 32

5. Discussion and Qualitative Comparison

Ghost tasking’s smaller latent space is linked to stronger inverse-problem data fit, while practical use trades algebraic and computational costs against theoretical guarantees. Compared with related GP methods, it aims to preserve strong task coupling without restricting systems to those already parametrizable.

5.1. Discussion of experimental results

  • Ghost tasking uses at most m additional tasks and n = p + m − q latent functions, ranging from m to q, while PIGP uses q latent functions.Fewer latent functions let a fixed amount of training data constrain each function more strongly, sharpening the data-fit term.Source: §5.1 · p. 34
  • In Experiment 3, GT+B’s rank-1 parametrization fits training data across five tasks, while PIGP’s rank-3 parametrization fits data across six tasks.PIGP’s latent functions are less constrained despite its larger total training set, producing a flatter data-fit term that does not fully counteract the complexity penalty.Source: §5.1 · p. 34

5.2. Practical considerations and limitations

  • Finding a suitable generator Λ is direct for holonomic systems in Macaulay2, but extension-module constructions are not currently applicable in the cited algebra programs.The authors therefore use a heuristic ansatz for Λ(x) and verify it computationally.Source: §5.2 · p. 34
  • Adding ghost-task collocation points requires no measurements, but computation scales cubically with training-data size and accuracy gains vary by experiment.The paper reports substantial gains for oscillatory systems but comparatively modest improvements elsewhere.Source: §5.2 · p. 34
  • For non-constant coefficients, parametrized kernels can become numerically ill-conditioned, potentially affecting stability in sharply peaked likelihood regimes [55].The paper points to stabilization and regularization techniques as possible mitigations.Source: §5.2 · p. 35
  • The torsion-free variant has lower algebraic and evaluation overhead than the projective variant, but does not guarantee parametrization of the full solution space.The authors recommend torsion-freeness in practice and report no practically relevant failures in their experiments.Source: §5.2 · p. 35

5.3. Qualitative Method Comparison

  • Analytical-kernel methods rely on known fundamental solutions, while parametrization-based methods are algorithmic but trade task count against coupling and applicability.Ghost tasking seeks a balance by relaxing parametrizability constraints while retaining stronger coupling; its broad theoretical scope is practically mediated by computer algebra implementations, largely limited to polynomial or rational coefficients.Source: §5.3 · pp. 35–36

6. Conclusion

Ghost tasking adds up to two auxiliary tasks to extend parametrizability while keeping the latent function space minimal. The conclusion reports comparable forward performance with fewer training data and frequent, several-orders-of-magnitude accuracy gains over PIGP on low-data inverse problems.

6. Conclusion

  • Ghost tasking adds one, and at most two, ghost tasks to extend parametrizability to systems that are not parametrizable on their own while retaining a minimal latent function space.The resulting space can fit all solutions of the system.Source: §6 · p. 36
  • On forward problems, ghost tasking performs comparably to PIGP with fewer training data and outperforms it when training-data amounts are equal.Source: §6 · p. 36
  • On inverse problems, ghost tasking frequently attains several orders of magnitude higher accuracy, particularly in the low-data regime.Source: §6 · p. 36
  • The conclusion anticipates that advances in constructive algebraic analysis and automated generator construction could broaden the systems ghost tasking can encode optimally.Source: §6 · p. 37

Appendix A. Computer algebra syntax

The appendix outlines computer-algebra procedures for constructing and checking ghost-task parametrizations. It also notes limits of available generator construction and parameter-specific certification.

Appendix A. Computer algebra syntax

  • The appendix provides Macaulay2 and OreModules syntax for constructing a ghost column Λ(x) and checking its validity.It refers readers to the authors’ GitHub page for full scripts.Source: Appendix A · p. 42
  • Certification over generic values of commuting constants can miss special values that break parametrizability, so the appendix advises substituting known numerical constants before certification.Source: Appendix A · p. 42
  • For holonomic systems, Macaulay2 can produce a cyclic generator algebraically; for extension modules, the authors use a loop over candidate operators and test each for projectivity or torsion-freeness.The loop stops at the first successful candidate and had worked reasonably well in practice.Source: Appendix A · p. 42
  • The appendix describes Ext-module checks for torsion-freeness and projectivity, including a right-inverse test for projectivity.It gives corresponding Macaulay2 and OreModules operations.Source: Appendix A · p. 43
  • Macaulay2 constructs a parametrization matrix from generators of the right null space, while OreModules provides a minimal-parametrization call.Macaulay2’s calculated parametrization need not have the minimal number of columns.Source: Appendix A · p. 43
  • In the measured f(t) setting, GT+ outperforms GT and PIGP for g/ℓ, while performance is similar in the known f(t) setting.The comparison aggregates results across 10 runs using geometric means; GT+ is unavailable for PIGP in the measured setting by definition.Source: Appendix A · p. 44

Appendix B. Ablation studies

The ablation tests whether extra collocation points on exactly known zero-valued tasks improve inverse parameter recovery. GT+ strongly improves recovery of g/ℓ in experiment 1, while the later experiments show weaker or absent benefits from the extra-point variants.

Appendix B. Ablation studies

  • The “+” variant adds collocation points for tasks known to be exactly zero, using a small noise nugget; results report median parameter errors and their geometric mean.The comparison spans standard and “+” variants in inverse settings across the experiments.Source: Appendix B · p. 44
  • GT+ strongly improves recovery of g/ℓ in experiment 1, a ratio related to the system's eigenfrequency.The authors suggest the standard GT method may absorb physically relevant oscillations into the ghost task, which GT+ can suppress.Source: Appendix B · p. 44
  • When f(t) is fully known and both methods receive 25 noiseless input points, GT and PIGP perform similarly and recover multimodal posteriors for ℓ and g.Figure B.14 shows the re-scaled log posteriors for both methods under this condition.Source: Appendix B · p. 44, Figure B.14 · p. 46
  • Experiment 2 shows no significant difference between standard and extra-point variants.The extra-point variants constrain the ghost task or inhomogeneities on 15 × 15 points.Source: Table B.2 · p. 45
  • GT outperforms PIGP in every setting of experiment 3, but the extra-point variants show no significant differences.The comparison uses 10 × 10 × 10 collocation points for the extra-point variants.Source: Table B.3 · p. 45

Appendix C. Application to systems with general non-constant coefficients

The appendix illustrates a commutative-polynomial-ring route to parametrizing systems with non-constant coefficients. Its validity requires g3(x) and g4(x) to be constant, limiting its use when those coefficients vary.

Appendix C. Application to systems with general non-constant coefficients

  • The appendix demonstrates a Macaulay2 parametrization route by treating general coefficient functions as implicitly constant parameters in commutative polynomial rings.The example uses a system with non-constant coefficients and a parametrization approach from prior work.Source: Appendix C · p. 46
  • A commutative-ring parametrization is valid only when g3(x) and g4(x) are constant; g1(x) and g2(x) have no such commutativity constraint.The validity condition follows from requiring g3(x) and g4(x) to commute with ∇2.Source: Appendix C · p. 46
  • The commutativity requirement rules out this approach for general applications where g3(x) and g4(x) depend on their arguments.When the operator-matrix elements commute, the approach offers a convenient, computationally inexpensive parametrization using standard commutative polynomial-ring tools.Source: Appendix C · p. 46
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