Physics-Informed Neural Networks (PINNs) have emerged as a powerful frame- work for solving partial differential equations (PDEs). However, standard MLP- based PINNs often fail to converge when dealing with complex initial value problems, leading to solutions that violate causality and suffer from a spectral bias towards low-frequency components. To address these issues, we introduce NeuSA (Neuro-Spectral Architectures), a novel class of PINNs inspired by classi- cal spectral methods, designed to solve linear and nonlinear PDEs with variable coefficients. NeuSA learns a projection of the underlying PDE onto a spectral basis, leading to a finite-dimensional representation of the dynamics which is then integrated with an adapted Neural ODE (NODE). This allows us to overcome spectral bias, by leveraging the high-frequency components enabled by the spectral representation; to enforce causality, by inheriting the causal structure of NODEs, and to start training near the target solution, by means of an initialization scheme based on classical methods. We validate NeuSA on canonical benchmarks for lin- ear and nonlinear wave equations, demonstrating strong performance as compared to other architectures, with faster convergence, improved temporal consistency and superior predictive accuracy. Code and pretrained models are available in https://github.com/arthur-bizzi/neusa.

Discovering similarities between online listings is a common backend task being used across different downstream experiences in eBay. Our baseline unstructured listing similarity method relies on measuring the semantic textual similarity between the embedding vectors of listing titles. However, we discovered that even with the latest contextualized embedding methods, our similarity fails to give the proper weight to the key tokens in the title that matter. This often results in identifying listing similarities that are not sufficient, which later hurts the downstream experiences. In this paper we present a method we call "Listing2Query", or "L2Q", which uses a Sequence Labeling approach to learn token importance from our users? search queries and on-site behaviour. We used pairs of listing titles and their matching search queries, and leveraged a contextualized character language model, to train L2Q as a bidirectional recurrent neural network to produce token importance weights. We demonstrate that plugging these weights into relatively straightforward listing similarity methods is a simple way to significantly improve the similarity results, even to the extent that it consistently outperforms those created by popular representations such as BERT. Notably, this approach is not reserved to only large online marketplaces but can be generalized to other cases that include a search-driven experience and a recall set of short documents.