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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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This content will become publicly available on June 1, 2026
An adaptive and stability-promoting layerwise training approach for sparse deep neural network architecture
This work presents a two-stage adaptive framework for progressively developing deep neural network (DNN) architectures that generalize well for a given training dataset. In the first stage, a layerwise training approach is adopted where a new layer is added each time and trained independently by freezing parameters in the previous layers. We impose desirable structures on the DNN by employing manifold regularization, sparsity regularization, and physics-informed terms. We introduce a $$\ epsilon-\delta$$ stability-promoting concept as a desirable property for a learning algorithm and show that employing manifold regularization yields a $$\epsilon-\delta$$ stability-promoting algorithm. Further, we also derive the necessary conditions for the trainability of a newly added layer and investigate the training saturation problem. In the second stage of the algorithm (post-processing), a sequence of shallow networks is employed to extract information from the residual produced in the first stage, thereby improving the prediction accuracy. Numerical investigations on prototype regression and classification problems demonstrate that the proposed approach can outperform fully connected DNNs of the same size. Moreover, by equipping the physics-informed neural network (PINN) with the proposed adaptive architecture strategy to solve partial differential equations, we numerically show that adaptive PINNs not only are superior to standard PINNs but also produce interpretable hidden layers with provable stability. We also apply our architecture design strategy to solve inverse problems governed by elliptic partial differential equations.
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- PAR ID:
- 10584603
- Publisher / Repository:
- CAMWA
- Date Published:
- Journal Name:
- Computer Methods in Applied Mechanics and Engineering
- Volume:
- 441
- Issue:
- C
- ISSN:
- 0045-7825
- Page Range / eLocation ID:
- 117938
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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