Scientific machine learning (SciML) explores the development of neural network models to approximate solutions to Partial Differential Equations (PDEs). However, there exists a significant research gap when computational domains are arbitrary manifolds, which are common in real-world scientific and engineering applications. The inherent challenge arises when calculating differential operators defined on curved surfaces, particularly in scenarios where surface parameterization is unavailable. In this paper, we present a neural network-based method for solving PDEs on surfaces described only by point clouds, without any other geometrical priors. Our method comprises two steps—local surface approximation based on graph neural networks and solving PDEs on point clouds. For surface reconstruction, our graph neural networks can be generalized based on the predictions of simple geometries during training to significantly more complicated surfaces for evaluation. The proposed approach demonstrates its capacity to learn geometric features from point cloud data without requiring external datasets, offers superior performance compared to benchmark models across various PDE types, and exhibits robustness in handling complex surfaces, non-uniform point distributions, and noise.

Differential equations are pivotal in modeling and understanding the dynamics of various systems, as they offer insights into their future states through parameter estimation fitted to time series data. In fields such as economy, politics, and biology, the observation data points in the time series are often independently obtained (i.e., Repeated Cross-Sectional (RCS) data). RCS data showed that traditional methods for parameter estimation in differential equations, such as using mean values of RCS data over time, Gaussian Process-based trajectory generation, and Bayesian-based methods, have limitations in estimating the shape of parameter distributions, leading to a significant loss of data information. To address this issue, this study proposes a novel method called Estimation of Parameter Distribution (EPD) that provides accurate distribution of parameters without loss of data information. EPD operates in three main steps: generating synthetic time trajectories by randomly selecting observed values at each time point, estimating parameters of a differential equation that minimizes the discrepancy between these trajectories and the true solution of the equation, and selecting the parameters depending on the scale of discrepancy. We then evaluated the performance of EPD across several models, including exponential growth, logistic population models, and target cell-limited models with delayed virus production, thereby demonstrating the ability of the proposed method in capturing the shape of parameter distributions. Furthermore, we applied EPD to real-world datasets, capturing various shapes of parameter distributions over a normal distribution. These results address the heterogeneity within systems, marking a substantial progression in accurately modeling systems using RCS data. Therefore, EPD marks a significant advancement in accurately modeling systems with RCS data, realizing a deeper understanding of system dynamics and parameter variability.

The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.

Learning graph-structured data with graph neural networks (GNNs) has been recently emerging as an important field because of its wide applicability in bioinformatics, chemoinformatics, social network analysis, and data mining. Recent GNN algorithms are based on neural message passing, which enables GNNs to integrate local structures and node features recursively. However, past GNN algorithms based on 1-hop neighborhood neural message passing are exposed to a risk of loss of information on local structures and relationships. In this article, we propose Neighborhood Edge AggregatoR (NEAR), a framework that aggregates relations between the nodes in the neighborhood via edges. NEAR, which can be orthogonally combined with Graph Isomorphism Network (GIN), gives integrated information that describes which nodes in the neighborhood are connected. Therefore, NEAR can reflect additional information of a local structure of each node beyond the nodes themselves in 1-hop neighborhood. Experimental results on multiple graph classification tasks show that our algorithm makes a good improvement over other existing 1-hop based GNN-based algorithms.

The convergence of generative adversarial networks (GANs) has been studied substantially in various aspects to achieve successful generative tasks. Ever since it is first proposed, the idea has achieved many theoretical improvements by injecting an instance noise, choosing different divergences, penalizing the discriminator, and so on. In essence, these efforts are to approximate a real-world measure with an idle measure through a learning procedure. In this article, we provide an analysis of GANs in the most general setting to reveal what, in essence, should be satisfied to achieve successful convergence. This work is not trivial since handling a converging sequence of an abstract measure requires a lot more sophisticated concepts. In doing so, we find an interesting fact that the discriminator can be penalized in a more general setting than what has been implemented. Furthermore, our experiment results substantiate our theoretical argument on various generative tasks.