Andrea Sacchetti

Physics-informed extreme learning machines (PIELMs) have emerged as optimisation-free surrogates to gradient-based physics-informed neural networks, yet their quantitative advantage over classical finite-element methods (FEM) has remained limited to regular-grid domains. This work delivers the first systematic, large-scale head-to-head study between a minimalist single- $R_{m}$ PIELM and FEM on four elliptic benchmarks of escalating difficulty with random collocation points: (i) a 2-D Poisson verification case, (ii) a 2-D heat-conduction problem with an internal cavity, (iii) a 2-D hyperbolic PDE featuring steep gradients, and (iv) the 3-D extension of (ii). More than $4 \times 10^{3}$ random hyper-parameter realisations are executed, spanning seven decades in training loss and error. For the two smooth 2-D benchmarks the PIELM attains $\approx 10^{-8}$ root-mean-square (RMS) errors while being up to an order of magnitude faster than equally accurate FEM. On the hyperbolic test it retains its speed edge and lowers the FEM error by one order of magnitude. In 3-D the PIELM reaches RMS values within a factor $\approx 4$ of the FEM optimum yet requires only a fraction of the wall-time, revealing dimensionality limits but preserving favourable accuracy-per-second scaling. Statistical analysis uncovers an almost linear log-log correlation between the collocation residual and the RMS error, enabling a practical a-posteriori stopping criterion. Sensitivity sweeps identify the hidden-layer width $M$ and the weight scale $R_{m}$ as the dominant hyper-parameters, whereas the interior-to-boundary collocation ratio primarily governs stability. Error visualisations expose complementary failure modes between PIELM (local spikes) and FEM (global diffusion), suggesting the need for more robust collocation-points selection strategies. Overall, the evidence positions the single- $R_{m}$ PIELM as a compelling alternative for fast, high-accuracy solutions of low- to moderate-dimensional, smooth PDEs, and charts a roadmap for overcoming the remaining challenges in high-dimensional and highly non-smooth regimes.

The ‘ball on a string’ demonstration is a common tool used in physics education to illustrate the concept of conservation of angular momentum. However, various confounding factors can cause significant deviations from the idealized case, particularly under extreme conditions or when using low-stiffness pivots or high coefficients of friction. These factors include air resistance, contact friction at the pivot point, the mass of the ball and string, the angle of the string due to gravity, and the wobbling of the pivot point due to the centrifugal forces acting on it. In this work, we critically review by means of accurate simulations the adequateness of the ‘ball on a string’ demonstration in view of these confounding factors and provide recommendations for instructors on how to maximize the educational value of the demonstration while minimizing potential confusion for students. Our analysis suggests that a stiff pivot and avoiding extreme conditions are key to obtaining results that are in good agreement with the idealized case. We also caution instructors against using the demonstration without at least mentioning the confounding factors, as this may lead to a questionable understanding of the underlying physics principles.

This work addresses a specific type of configurations where typical-tracing simulations can fail to produce meaningful results, namely when the light intensity transmitted by a very small aperture needs to be determined. In such cases it is indeed computational very expensive, if not prohibitive, to simulate all the rays that are needed in order to get a statistically significant number through the aperture. Our proposed solution exploits a virtual source to intrinsically limit the simulation only to those rays that do cross the aperture thus massively enhancing the computational efficiency. We lay down the mathematical formalism behind the method and present it also in the form of a simple generalizable algorithm. Finally, we provide an experimental test that confirms the validity of our proposal over a range of aperture sizes.

We compare the Finite Element Method (FEM) simulation of a standard Partial Differential Equation thermal problem of a plate with a hole with a Neural Network (NN) simulation. The largest deviation from the true solution obtained from FEM (0.015 for a solution on the order of unity) is easily achieved with NN too without much tuning of the hyperparameters. Accuracies below 0.01 instead require refinement with an alternative optimizer to reach a similar performance with NN. A rough comparison between the Floating Point Operations values, as a machine-independent quantification of the computational performance, suggests a significant difference between FEM and NN in favour of the former. This also strongly holds for computation time: for an accuracy on the order of 10 ⁻⁵ , FEM and NN require 54 and 1100 seconds, respectively. A detailed analysis of the effect of varying different hyperparameters shows that accuracy and computational time only weakly depend on the major part of them. Accuracies below 0.01 cannot be achieved with the “adam” optimizers and it looks as though accuracies below 10 ⁻⁵ cannot be achieved at all. In conclusion, the present work shows that for the concrete case of solving a steady-state 2D heat equation, the performance of a FEM algorithm is significantly better than the solution via networks.