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Weak forms offer strong regularisations: how to make physics-informed (quantum) machine learning more robust

Annie E. Paine, Smit Chaudhary, Antonio A. Gentile

TL;DR

The paper tackles robustness gaps in physics-informed differential equation solvers that rely on pointwise residuals, which can fail to propagate boundary information and generalize. It proposes a hybrid loss that couples a local collocation term with a global weak-form regularization, leveraging integration by parts and test functions, and shows this approach works well with differentiable quantum architectures under domain decomposition. Across damped oscillators, Burgers, a linear 2D problem, and Laplace equations, the hybrid method outperforms purely collocation or purely weak formulations, reducing susceptibility to trivial solutions and improving convergence. This approach offers a practical pathway to more robust, domain-agnostic PI solvers on near-term quantum hardware, with clear guidance for loss design and domain-partitioned training.

Abstract

Physics-informed (PI) methodologies have surged to become a pillar route to solve Differential Equations (DEs), sustained by the growth of machine learning methods in scientific contexts. The main proposition of PI is to minimise variationally a loss function, formally ensuring that a neural surrogate of the solution has the DE locally satisfied. The nature of such formulation encouraged the exploration of equivalent quantum algorithms, where the surrogate solution is expressed by variational quantum architectures. The locality of typical loss functions emphasises the DE to hold at an ensemble of points sampled in the domain, but encounters issues when generalising beyond such points, or when propagating boundary conditions. Issues which affect classical and quantum PI algorithms alike. The quest to fill this gap in robustness and accuracy against mainstream DE solvers has led to a plethora of proposals in various directions. In particular, classical DE solvers have long employed the weak form - an integral based approach aiming at imposing a global condition on the solution - prioritising a good average behaviour instead of ``overfitting'' select points. Here, we propose and explore to combine contributions from both local and global loss functions in PI routines, to exploit the advantages and mitigate the weaknesses of both. We showcase this intuition in a variety of problems focusing on differentiable quantum architectures, and demonstrating in particular how orchestrating such hybrid loss formulation with domain decomposition can offer a strong advantage over local-only strategies.

Weak forms offer strong regularisations: how to make physics-informed (quantum) machine learning more robust

TL;DR

The paper tackles robustness gaps in physics-informed differential equation solvers that rely on pointwise residuals, which can fail to propagate boundary information and generalize. It proposes a hybrid loss that couples a local collocation term with a global weak-form regularization, leveraging integration by parts and test functions, and shows this approach works well with differentiable quantum architectures under domain decomposition. Across damped oscillators, Burgers, a linear 2D problem, and Laplace equations, the hybrid method outperforms purely collocation or purely weak formulations, reducing susceptibility to trivial solutions and improving convergence. This approach offers a practical pathway to more robust, domain-agnostic PI solvers on near-term quantum hardware, with clear guidance for loss design and domain-partitioned training.

Abstract

Physics-informed (PI) methodologies have surged to become a pillar route to solve Differential Equations (DEs), sustained by the growth of machine learning methods in scientific contexts. The main proposition of PI is to minimise variationally a loss function, formally ensuring that a neural surrogate of the solution has the DE locally satisfied. The nature of such formulation encouraged the exploration of equivalent quantum algorithms, where the surrogate solution is expressed by variational quantum architectures. The locality of typical loss functions emphasises the DE to hold at an ensemble of points sampled in the domain, but encounters issues when generalising beyond such points, or when propagating boundary conditions. Issues which affect classical and quantum PI algorithms alike. The quest to fill this gap in robustness and accuracy against mainstream DE solvers has led to a plethora of proposals in various directions. In particular, classical DE solvers have long employed the weak form - an integral based approach aiming at imposing a global condition on the solution - prioritising a good average behaviour instead of ``overfitting'' select points. Here, we propose and explore to combine contributions from both local and global loss functions in PI routines, to exploit the advantages and mitigate the weaknesses of both. We showcase this intuition in a variety of problems focusing on differentiable quantum architectures, and demonstrating in particular how orchestrating such hybrid loss formulation with domain decomposition can offer a strong advantage over local-only strategies.
Paper Structure (23 sections, 25 equations, 4 figures)

This paper contains 23 sections, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Formulations of a Differential Equation.
  4. (Quantum) physics-informed methods.
  5. Trainable models
  6. Loss function
  7. The Weak Form as a regularization term
  8. Combining loss contributions.
  9. Domain Decomposition
  10. Training and its cost.
  11. Results
  12. Damped Oscillator
  13. Stationary Burgers Equation
  14. Linear 2D Example
  15. Laplace
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Attained solution (a) and training (b) of Eq. \ref{['eq:damped_osc']}, comparing the baseline analytical solution (truth) against the converged solutions calculated adopting as the relevant loss term(s) to train against: (coll) only the collocation-based Eq. \ref{['eq:loss_residual']}, (coll join) the latter, inclusive of $\mathcal{L}_{\mathrm{SBC}}$ from Eq. \ref{['eq:domdecomp_lossterm']}, (weak) the weak-formulation alone or (both) a combination of both weak and collocation contributions, as seen in Eq. \ref{['eq:combination']}. The evolution of the latter is shown in (b) as a solid line, against training epoch and along with (as dashed line) the measure-of-success metric as defined in the main text.
  • Figure 2: Attained solution (a) and training (b) of stationary Burgers Eq.\ref{['eq:stnry_burgers']}. In (a) we report the baseline analytical solution (truth) as a solid line, against the converged solutions adopting various strategies as dashed lines, labelled according to the caption of Fig. \ref{['fig:damped_osc']}, to provide the relevant loss function for the training. The evolution of the latter is shown in (b) as a solid line, against training epoch and along with (as dashed line) the measure-of-success metric as defined in the main text.
  • Figure 3: Attained solution (left) and training (right) of linear 2D Eq.\ref{['eq:simple_2D']}. In the 3D plots we report the baseline analytical solution (True) against the converged solutions adopting various strategies, labelled according to the caption of Fig. \ref{['fig:damped_osc']}, to provide the relevant loss function for the training. The evolution of the latter is shown in (b) as a solid line (l), against training epoch and along with (as dashed line) the measure-of-success metric (m) as defined in the main text. The 3D plots displaying the trained solutions colour-code the difference against the baseline solution (colour-coded against the value of $f(x,y)$ at each point.
  • Figure 4: Attained solution (left) and training (right) of Laplace Eq.\ref{['eq:laplace']}. In the 3D plots we report the baseline analytical solution (True) against the converged solutions adopting various strategies, labelled according to the caption of Fig. \ref{['fig:damped_osc']}, to provide the relevant loss function for the training. The evolution of the latter is shown in (b) as a solid line (l), against training epoch and along with (as dashed line) the measure-of-success metric (m) as defined in the main text. The 3D plots displaying the trained solutions colour-code the difference against the baseline solution (colour-coded against the value of $f(x,y)$ at each point.