Ísak Pétursson

Ísak Pétursson, María Óskarsdóttir

Stochastic partial differential equations (SPDEs) describe the evolution of random processes over space and time, but their solutions are often analytically intractable and computationally expensive to estimate. In this paper, we propose the Learned Expectation Collapser (LEC), a physics-informed neural framework designed to approximate the expected value of linear SPDE solutions without requiring domain discretization. By leveraging randomized sampling of both space-time coordinates and noise realizations during training, LEC trains standard feedforward neural networks to minimize residual loss across multiple stochastic samples. We hypothesize and empirically confirm that this training regime drives the network to converge toward the expected value of the solution of the SPDE. Using the stochastic heat equation as a testbed, we evaluate performance across a diverse set of 144 experimental configurations that span multiple spatial dimensions, noise models, and forcing functions. The results show that the model consistently learns accurate approximations of the expected value of the solution in lower dimensions and a predictable decrease in accuracy with increased spatial dimensions, with improved stability and robustness under increased Monte Carlo sampling. Our findings offer new insight into how neural networks implicitly learn statistical structure from stochastic differential operators and suggest a pathway toward scalable, simulator-free SPDE solvers.

Ísak Pétursson, María Óskarsdóttir

We present a novel approach for predicting the distribution of asset returns using a quantile-based method with Long Short-Term Memory (LSTM) networks. Our model is designed in two stages: the first focuses on predicting the quantiles of normalized asset returns using asset-specific features, while the second stage incorporates market data to adjust these predictions for broader economic conditions. This results in a generalized model that can be applied across various asset classes, including commodities, cryptocurrencies, as well as synthetic datasets. The predicted quantiles are then converted into full probability distributions through kernel density estimation, allowing for more precise return distribution predictions and inferencing. The LSTM model significantly outperforms a linear quantile regression baseline by 98% and a dense neural network model by over 50%, showcasing its ability to capture complex patterns in financial return distributions across both synthetic and real-world data. By using exclusively asset-class-neutral features, our model achieves robust, generalizable results.