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Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations (PDEs) in a forward and inverse manner using neural networks. However, balancing individual loss terms can be challenging, mainly when training these networks for stiff PDEs and scenarios requiring enforcement of numerous constraints. Even though statistical methods can be applied to assign relative weights to the regression loss for data, assigning relative weights to equation-based loss terms remains a formidable task. This paper proposes a method for assigning relative weights to the mean squared loss terms in the objective function used to train PINNs. Due to the presence of temporal gradients in the governing equation, the physics-informed loss can be recast using numerical integration through backward Euler discretization. The physics-uninformed and physics-informed networks should yield identical predictions when assessed at corresponding spatiotemporal positions. We refer to this consistency as “temporal consistency.” This approach introduces a unique method for training physics-informed neural networks (PINNs), redefining the loss function to allow for assigning relative weights with statistical properties of the observed data. In this work, we consider the two- and three-dimensional Navier–Stokes equations and determine the kinematic viscosity using the spatiotemporal data on the velocity and pressure fields. We consider numerical datasets to test our method. We test the sensitivity of our method to the timestep size, the number of timesteps, noise in the data, and spatial resolution. Finally, we use the velocity field obtained using particle image velocimetry experiments to generate a reference pressure field and test our framework using the velocity and pressure fields.
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SenseNet: A Physics-Informed Deep Learning Model for Shape Sensing
Shape sensing is an emerging technique for the reconstruction of deformed shapes using data from a discrete network of strain sensors. The prominence is due to its suitability in promising applications such as structural health monitoring in multiple engineering fields and shape capturing in the medical field. In this work, a physics-informed deep learning model, named SenseNet, was developed for shape sensing applications. Unlike existing neural network approaches for shape sensing, SenseNet incorporates the knowledge of the physics of the problem, so its performance does not rely on the choices of the training data. Compared with numerical physics-based approaches, SenseNet is a mesh-free method, and therefore it offers convenience to problems with complex geometries. SenseNet is composed of two parts: a neural network to predict displacements at the given input coordinates, and a physics part to compute the loss using a function incorporated with physics information. The prior knowledge considered in the loss function includes the boundary conditions and physics relations such as the strain–displacement relation, material constitutive equation, and the governing equation obtained from the law of balance of linear momentum.SenseNet was validated with finite-element solutions for cases with nonlinear displacement fields and stress fields using bending and fixed tension tests, respectively, in both two and three dimensions. A study of the sensor density effects illustrated the fact that the accuracy of the model can be improved using a larger amount of strain data. Because general three dimensional governing equations are incorporated in the model, it was found that SenseNet is capable of reconstructing deformations in volumes with reasonable accuracy using just the surface strain data. Hence, unlike most existing models, SenseNet is not specialized for certain types of elements, and can be extended universally for even thick-body applications.
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- Award ID(s):
- 1911836
- PAR ID:
- 10494469
- Editor(s):
- -
- Publisher / Repository:
- Journal of Engineering Mechanics
- Date Published:
- Journal Name:
- Journal of Engineering Mechanics
- Edition / Version:
- -
- Volume:
- 149
- Issue:
- 3
- ISSN:
- 0733-9399
- Subject(s) / Keyword(s):
- -
- Format(s):
- Medium: X Size: - Other: -
- Size(s):
- -
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1061/JENMDT.EMENG-6901
