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Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet) to identify candidate functions on which to expand the solution of PDEs. We have devised an approach that uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions that have the following properties: (1) they constitute a basis, (2) they are orthonormal, and (3) they are hierarchical, i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs. The proposed approach advances the state of the art and versatility of spectral methods and, more generally, promotes the synergy between traditional scientific computing and machine learning. [ABSTRACT FROM AUTHOR]

Machine Learning methods are emerging as faster and efficient alternatives to numerical simulation techniques. The field of Scientific Computing has started adopting these data-driven approaches to faithfully model physical phenomena using scattered, noisy observations from coarse-grained grid-based simulations. In this paper, we investigate data-driven modelling of Bose-Einstein Condensates (BECs). In particular, we use Gaussian Processes (GPs) to model the ground state wave function of BECs as a function of scattering parameters from the dimensionless Gross Pitaveskii Equation (GPE). Experimental results illustrate the ability of GPs to accurately reproduce ground state wave functions using a limited number of data points from simulations. Consistent performance across different configurations of BECs, namely Scalar and Vectorial BECs generated under different potentials, including harmonic, double well and optical lattice potentials pronounces the versatility of our method. Comparison with existing data-driven models indicates that our model achieves similar accuracy with only a small fraction ( 1 50 th) of data points used by existing methods, in addition to modelling uncertainty from data. When used as a simulator post-training, our model generates ground state wave functions 36 × faster than Trotter Suzuki, a numerical approximation technique that uses Imaginary time evolution. Our method is quite general; with minor changes it can be applied to similar quantum many-body problems. [ABSTRACT FROM AUTHOR]

Transient simulations of dynamic systems, using physics-based scientific computing tools, are practically limited by availability of computational resources and power. While the promise of machine learning has been explored in a variety of scientific disciplines, its application in creation of a framework for computationally expensive transient models has not been fully explored. Here, we present an ensemble approach where one such computationally expensive tool, discrete element method, is combined with time-series forecasting via auto regressive integrated moving average and machine learning methods to simulate a complex pharmaceutical problem: development of an agitation protocol in an agitated filter dryer to ensure uniform solid bed mixing. This ensemble approach leads to a significant reduction in the computational burden, while retaining model accuracy and performance, practically rendering simulations possible. The developed machine-learning model shows good predictability and agreement with the literature, demonstrating its tremendous potential in scientific computing. [ABSTRACT FROM AUTHOR]