Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often necessary to compute the solutions or a function of the solutions to a parametric PDE in multiple scenarios corresponding to different parameters of this PDE. This process often requires resolving the PDEs from scratch, which is time-consuming. To better employ existing simulations for the PDEs, we propose a framework for finding solutions to parabolic PDEs across different scenarios by meta-learning an underlying base distribution. We build upon this base distribution to propose a method for computing solutions to parametric PDEs under different parameter settings. Finally, we illustrate the application of the proposed methods through extensive experiments in generative modeling, stochastic control, and finance. The empirical results suggest that the proposed approach improves generalization to solving PDEs under new parameter regimes.

Despite the rapid and significant advancements in deep learning for Quantitative Structure-Activity Relationship (QSAR) models, the challenge of learning robust molecular representations that effectively generalize in real-world scenarios to novel compounds remains an elusive and unresolved task. This study examines how atom-level pretraining with quantum mechanics (QM) data can mitigate violations of assumptions regarding the distributional similarity between training and test data and therefore improve performance and generalization in downstream tasks. In the public dataset Therapeutics Data Commons (TDC), we show how pretraining on atom-level QM improves performance overall and makes the activation of the features distributes more Gaussian-like which results in a representation that is more robust to distribution shifts. To the best of our knowledge, this is the first time that hidden state molecular representations are analyzed to compare the effects of molecule-level and atom-level pretraining on QM data.