A neural network-based PDE solving algorithm with high precision

Jiang, Zichao; Jiang, Junyang; Yao, Qinghe; Yang, Guochen

doi:10.1038/s41598-023-31236-0

Cited by 8 publications

(2 citation statements)

References 38 publications

(49 reference statements)

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“…In practice, data-driven models are trained on numerical simulation results and approximate a solution to the system of equations. The inference step of the successfully trained model takes a fraction of the computational resources compared to the full mechanistic model ( 63 , 64 ).…”

Section: Facets Of Mechanistic Learningmentioning

confidence: 99%

A review of mechanistic learning in mathematical oncology

Metzcar,

Jutzeler,

Macklin

et al. 2024

Front. Immunol.

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Mechanistic learning refers to the synergistic combination of mechanistic mathematical modeling and data-driven machine or deep learning. This emerging field finds increasing applications in (mathematical) oncology. This review aims to capture the current state of the field and provides a perspective on how mechanistic learning may progress in the oncology domain. We highlight the synergistic potential of mechanistic learning and point out similarities and differences between purely data-driven and mechanistic approaches concerning model complexity, data requirements, outputs generated, and interpretability of the algorithms and their results. Four categories of mechanistic learning (sequential, parallel, extrinsic, intrinsic) of mechanistic learning are presented with specific examples. We discuss a range of techniques including physics-informed neural networks, surrogate model learning, and digital twins. Example applications address complex problems predominantly from the domain of oncology research such as longitudinal tumor response predictions or time-to-event modeling. As the field of mechanistic learning advances, we aim for this review and proposed categorization framework to foster additional collaboration between the data- and knowledge-driven modeling fields. Further collaboration will help address difficult issues in oncology such as limited data availability, requirements of model transparency, and complex input data which are embraced in a mechanistic learning framework

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Section: Facets Of Mechanistic Learningmentioning

confidence: 99%

A review of mechanistic learning in mathematical oncology

Metzcar,

Jutzeler,

Macklin

et al. 2024

Front. Immunol.

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“…The use of PINNs is currently being studied as a potential replacement for existing numerical techniques. Due to the recent advent of this type of neural networks, the literature is not yet massive, but reports particularly important pivotal works such as e.g [3,4,5,6,7,8,9,10,11,12,13,14,15,16]. A comprehensive review can be also found in [17].…”

Section: Introductionmentioning

confidence: 99%

Learning systems of ordinary differential equations with Physics-Informed Neural Networks: the case study of enzyme kinetics

Lecca

2024

J. Phys.: Conf. Ser.

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Physics Informed Neural Networks (PINNs) are a type of function approximators that use both data-driven supervised neural networks to learn the model of the dynamics of a physical system, and mathematical equations of the physical laws governing that system. PINNs have the benefit of being data-driven to train a model, but also of being able to assure consistency with the physics, and to extrapolate accurately beyond the range of data that currently accessible. As a result, PINNs can provide models that are more reliable while using less data. Specifically, the PINNs objective is to learn the solutions of a systems of equations using supervised learning on the available data and incorporating the knowledge of physical laws and constraints into the training process. However, solving single differential equations with a PINN may be relatively simple, solving systems of coupled differential equations may not be so simple. In this study, I present a neural network model specialized in solving differential equations of enzyme kinetics that has the main characteristic of being a demonstrative simple case of coupled equations system. The study focuses mainly on the theoretical aspects of the definition of a physics-informed loss function and shows a case study that highlights the challenges still to be overcome in solving systems of coupled differential equations.

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