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Physics-informed neural networks (PINNs) constitute a flexible approach to both finding solutions and identifying parameters of partial differential equations. Most works on the topic assume noiseless data, or data contaminated with weak Gaussian noise. We show that the standard PINN framework breaks down in case of non-Gaussian noise. We give a way of resolving this fundamental issue and we propose to jointly train an energy-based model (EBM) to learn the correct noise distribution. We illustrate the improved performance of our approach using multiple examples.

Physics-informed neural networks (PINNs) have proven an effective tool for solving differential equations, in particular when considering non-standard or ill-posed settings. When inferring solutions and parameters of the differential equation from data, uncertainty estimates are preferable to point estimates, as they give an idea about the accuracy of the solution. In this work, we consider the inverse problem and employ repulsive ensembles of PINNs (RE-PINN) for obtaining such estimates. The repulsion is implemented by adding a particular repulsive term to the loss function, which has the property that the ensemble predictions correspond to the true Bayesian posterior in the limit of infinite ensemble members. Where possible, we compare the ensemble predictions to Monte Carlo baselines. Whereas the standard ensemble tends to collapse to maximum-a-posteriori solutions, the repulsive ensemble produces significantly more accurate uncertainty estimates and exhibits higher sample diversity.

Machine learning models can be improved by adapting them to respect existing background knowledge. In this paper we consider multitask Gaussian processes, with background knowledge in the form of constraints that require a specific sum of the outputs to be constant. This is achieved by conditioning the prior distribution on the constraint fulfillment. The approach allows for both linear and nonlinear constraints. We demonstrate that the constraints are fulfilled with high precision and that the construction can improve the overall prediction accuracy as compared to the standard Gaussian process.

Generative adversarial networks constitute a powerful approach to generative modeling. While generated samples often are indistinguishable from real data, there is no guarantee that they will follow the true data distribution. For scientific applications in particular, it is essential that the true distribution is well captured by the generated distribution. In this work, we propose a method to ensure that the distributions of certain generated data statistics coincide with the respective distributions of the real data. In order to achieve this, we add a new loss term to the generator loss function, which quantifies the difference between these distributions via suitable f-divergences. Kernel density estimation is employed to obtain representations of the true distributions, and to estimate the corresponding generated distributions from minibatch values at each iteration. When compared to other methods, our approach has the advantage that the complete shapes of the distributions are taken into account. We evaluate the method on a synthetic dataset and a real-world dataset and demonstrate improved performance of our approach.

The planned IceCube-Gen2 radio neutrino detector at the South Pole will enhance the detection of cosmic ultra-high-energy neutrinos. It is crucial to utilize the available time until construction to optimize the detector design. A fully differentiable pipeline, from signal generation to detector response, would allow for the application of gradient descent techniques to explore the parameter space of the detector. In our work, we focus on the aspect of signal generation, and propose a modularized deep learning architecture to generate radio signals from in-ice neutrino interactions conditioned on the shower energy and viewing angle. The model is capable of generating differentiable signals with amplitudes spanning multiple orders of magnitude, as well as consistently producing signals corresponding to the same underlying event for different viewing angles. The modularized approach ensures physical consistency of the samples and leads to advantageous computational properties when using the model as part of a bigger optimization pipeline.