We investigate methods for learning partial differential equation (PDE) models from spatio-temporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyse the performance in using previous methods to denoise data for the task of discovering the governing system of PDEs. We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e. the advection–diffusion, classical Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) and nonlinear Fisher–KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model.
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Biologically-informed neural networks guide mechanistic modeling from sparse experimental data
Biologically-informed neural networks (BINNs), an extension of physics-informed neural networks [1], are introduced and used to discover the underlying dynamics of biological systems from sparse experimental data. In the present work, BINNs are trained in a supervised learning framework to approximate in vitro cell biology assay experiments while respecting a generalized form of the governing reaction-diffusion partial differential equation (PDE). By allowing the diffusion and reaction terms to be multilayer perceptrons (MLPs), the nonlinear forms of these terms can be learned while simultaneously converging to the solution of the governing PDE. Further, the trained MLPs are used to guide the selection of biologically interpretable mechanistic forms of the PDE terms which provides new insights into the biological and physical mechanisms that govern the dynamics of the observed system. The method is evaluated on sparse real-world data from wound healing assays with varying initial cell densities [2].
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- Award ID(s):
- 1838314
- NSF-PAR ID:
- 10302339
- Editor(s):
- Lavrik, Inna
- Date Published:
- Journal Name:
- PLOS Computational Biology
- Volume:
- 16
- Issue:
- 12
- ISSN:
- 1553-7358
- Page Range / eLocation ID:
- e1008462
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by “baking” the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers’, and Navier-Stokes equations.
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Purpose To develop a method of biologically guided deep learning for post-radiation 18 FDG-PET image outcome prediction based on pre-radiation images and radiotherapy dose information. Methods Based on the classic reaction–diffusion mechanism, a novel biological model was proposed using a partial differential equation that incorporates spatial radiation dose distribution as a patient-specific treatment information variable. A 7-layer encoder–decoder-based convolutional neural network (CNN) was designed and trained to learn the proposed biological model. As such, the model could generate post-radiation 18 FDG-PET image outcome predictions with breakdown biological components for enhanced explainability. The proposed method was developed using 64 oropharyngeal patients with paired 18 FDG-PET studies before and after 20-Gy delivery (2 Gy/day fraction) by intensity-modulated radiotherapy (IMRT). In a two-branch deep learning execution, the proposed CNN learns specific terms in the biological model from paired 18 FDG-PET images and spatial dose distribution in one branch, and the biological model generates post-20-Gy 18 FDG-PET image prediction in the other branch. As in 2D execution, 718/233/230 axial slices from 38/13/13 patients were used for training/validation/independent test. The prediction image results in test cases were compared with the ground-truth results quantitatively. Results The proposed method successfully generated post-20-Gy 18 FDG-PET image outcome prediction with breakdown illustrations of biological model components. Standardized uptake value (SUV) mean values in 18 FDG high-uptake regions of predicted images (2.45 ± 0.25) were similar to ground-truth results (2.51 ± 0.33). In 2D-based Gamma analysis, the median/mean Gamma Index (<1) passing rate of test images was 96.5%/92.8% using the 5%/5 mm criterion; such result was improved to 99.9%/99.6% when 10%/10 mm was adopted. Conclusion The developed biologically guided deep learning method achieved post-20-Gy 18 FDG-PET image outcome predictions in good agreement with ground-truth results. With the breakdown biological modeling components, the outcome image predictions could be used in adaptive radiotherapy decision-making to optimize personalized plans for the best outcome in the future.more » « less
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: In order to evaluate urban earthquake resilience, reliable structural modeling is needed. However, detailed modeling of a large number of structures and carrying out time history analyses for sets of ground motions are not practical at an urban scale. Reduced-order surrogate models can expedite numerical simulations while maintaining necessary engineering accuracy. Neural networks have been shown to be a powerful tool for developing surrogate models, which often outperform classical surrogate models in terms of scalability of complex models. Training a reliable deep learning model, however, requires an immense amount of data that contain a rich input-output relationship, which typically cannot be satisfied in practical applications. In this paper, we propose model-informed symbolic neural networks (MiSNN) that can discover the underlying closed-form formulations (differential equations) for a reduced-order surrogate model. The MiSNN will be trained on datasets obtained from dynamic analyses of detailed reinforced concrete special moment frames designed for San Francisco, California, subject to a series of selected ground motions. Training the MiSNN is equivalent to finding the solution to a sparse optimization problem, which is solved by the Adam optimizer. The earthquake ground acceleration and story displacement, velocity, and acceleration time histories will be used to train 1) an integrated SNN, which takes displacement and velocity states and outputs the absolute acceleration response of the structure; and 2) a distributed SNN, which distills the underlying equation of motion for each story. The results show that the MiSNN can reduce computational cost while maintaining high prediction accuracy of building responses.more » « less
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Partial differential equations are common models in biology for predicting and explaining complex behaviors. Nevertheless, deriving the equations and estimating the corresponding parameters remains challenging from data. In particular, the fine description of the interactions between species requires care for taking into account various regimes such as saturation effects. We apply a method based on neural networks to discover the underlying PDE systems, which involve fractional terms and may also contain integration terms based on observed data. Our proposed framework, called Frac-PDE-Net, adapts the PDE-Net 2.0 by adding layers that are designed to learn fractional and integration terms. The key technical challenge of this task is the identifiability issue. More precisely, one needs to identify the main terms and combine similar terms among a huge number of candidates in fractional form generated by the neural network scheme due to the division operation. In order to overcome this barrier, we set up certain assumptions according to realistic biological behavior. Additionally, we use an L2-norm based term selection criterion and the sparse regression to obtain a parsimonious model. It turns out that the method of Frac-PDE-Net is capable of recovering the main terms with accurate coefficients, allowing for effective long term prediction. We demonstrate the interest of the method on a biological PDE model proposed to study the pollen tube growth problem.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1371/journal.pcbi.1008462
