Michael Triantafyllou

Khemraj Shukla, Zongren Zou, Theo Kaeufer et al.

Physics-informed neural networks (PINNs) have emerged as a promising framework for solving inverse problems governed by partial differential equations (PDEs), including the reconstruction of turbulent flow fields from sparse data. However, most existing PINN formulations are deterministic and do not provide reliable quantification of epistemic uncertainty, which is critical for ill-posed problems such as data-driven Reynolds-averaged Navier-Stokes (RANS) modeling. In this work, we develop and systematically evaluate a set of probabilistic extensions of PINNs for uncertainty quantification in turbulence modeling. The proposed framework combines (i) Bayesian PINNs with Hamiltonian Monte Carlo sampling and a tempered multi-component likelihood, (ii) Monte Carlo dropout, and (iii) repulsive deep ensembles that enforce diversity in function space. Particular emphasis is placed on the role of ensemble diversity and likelihood tempering in improving uncertainty calibration for PDE-constrained inverse problems. The methods are assessed on a hierarchy of test cases, including the Van der Pol oscillator and turbulent flow past a circular cylinder at Reynolds numbers Re=3,900 (direct numerical simulation data) and Re = 10,000 (experimental particle image velocimetry data). The results demonstrate that Bayesian PINNs provide the most consistent uncertainty estimates across all inferred quantities, while function-space repulsive ensembles offer a computationally efficient approximation with competitive accuracy for primary flow variables. These findings provide quantitative insight into the trade-offs between accuracy, computational cost, and uncertainty calibration in physics-informed learning, and offer practical guidance for uncertainty quantification in data-driven turbulence modeling.

Qianying Cao, Shanqing Liu, Alan John Varghese et al.

Time series forecasting plays a pivotal role in a wide range of applications, including weather prediction, healthcare, structural health monitoring, predictive maintenance, energy systems, and financial markets. While models such as LSTM, GRU, Transformers, and State-Space Models (SSMs) have become standard tools in this domain, selecting the optimal architecture remains a challenge. Performance comparisons often depend on evaluation metrics and the datasets under analysis, making the choice of a universally optimal model controversial. In this work, we introduce a flexible automated framework for time series forecasting that systematically designs and evaluates diverse network architectures by integrating LSTM, GRU, multi-head Attention, and SSM blocks. Using a multi-objective optimization approach, our framework determines the number, sequence, and combination of blocks to align with specific requirements and evaluation objectives. From the resulting Pareto-optimal architectures, the best model for a given context is selected via a user-defined preference function. We validate our framework across four distinct real-world applications. Results show that a single-layer GRU or LSTM is usually optimal when minimizing training time alone. However, when maximizing accuracy or balancing multiple objectives, the best architectures are often composite designs incorporating multiple block types in specific configurations. By employing a weighted preference function, users can resolve trade-offs between objectives, revealing novel, context-specific optimal architectures. Our findings underscore that no single neural architecture is universally optimal for time series forecasting. Instead, the best-performing model emerges as a data-driven composite architecture tailored to user-defined criteria and evaluation objectives.

José del Águila Ferrandis, Michael Triantafyllou, Chryssostomos Chryssostomidis et al.

Predicting motions of vessels in extreme sea states represents one of the most challenging problems in naval hydrodynamics. It involves computing complex nonlinear wave-body interactions, hence taxing heavily computational resources. Here, we put forward a new simulation paradigm by training recurrent type neural networks (RNNs) that take as input the stochastic wave elevation at a certain sea state and output the main vessel motions, e.g., pitch, heave and roll. We first compare the performance of standard RNNs versus GRU and LSTM neural networks (NNs) and show that LSTM NNs lead to the best performance. We then examine the testing error of two representative vessels, a catamaran in sea state 1 and a battleship in sea state 8. We demonstrate that good accuracy is achieved for both cases in predicting the vessel motions for unseen wave elevations. We train the NNs with expensive CFD simulations offline, but upon training, the prediction of the vessel dynamics online can be obtained at a fraction of a second. This work is motivated by the universal approximation theorem for functionals [1], and it is the first implementation of such theory to realistic engineering problems.