Baking physics into deep learning for modeling scientific problems

3D concrete printing (3DCP) is an innovative construction method that extrudes cementitious materials layer-by-layer to fabricate building components based on a digital model. 3DCP has gained increasing adoption globally for projects like buildings, bridges, retaining walls, and stormwater management systems. However, 3DCP places stringent demands on the rheological properties of the printable cementitious materials. Rheology is critical for the flow and buildability of fresh concrete during extrusion. The mixture must have a low yield stress for pumping, moderate viscosity to hold its shape after deposition, minimal bleeding and segregation of aggregates, as well as responsive rheology that can transition from fluid to solid state as layers are printed sequentially. In this dissertation, we aim to solve the challenges mentioned above including: (1) accurate and efficient quantification of rheological and thixotropic properties of cementitious materials, (2) evaluation of flow behavior of fresh concrete during extrusion. Using traditional experimental methods to evaluate rheological properties of cement paste is labor-intensive and time-consuming. Numerical simulations like Finite Element Method (FEM) or Computational Fluid Dynamics (CFD) can help increase the speed of finding the most suitable material for 3DCP. However, these simulation methods are costly in computing power, making them less desirable for simulating the dynamic and complex process of 3DCP. Moreover, mesh-based approaches are not suitable for complex geometries as the mesh generation would be challenging with curved boundaries, small features, holes or thin regions. Physics-Informed Neural Network (PINN) is introduced in the work as an innovative solution to simulate and model the whole process of 3DCP. It can leverage the flexibility and computational efficiency of data-based neural networks, which can accelerate the learning and convergence speed. At the same time, Partial Differential Equations (PDEs) would be embedded in the network structure to increase the accuracy of its prediction. The integrated governing physical laws can help the neural network understanding the underlying physics of the 3DCP process, potentially increasing the accuracy and reliability of simulations. A significant part of the research is dedicated to understanding the rheological and thixotropic behavior of cementitious materials, which is critical for the material design in 3DCP. We developed a Rheology-informed Neural Network, RheologyNet, where we embedded the rheological constitutive laws into the loss function. These physical laws can help the network focus on simulating the rheological properties of cementitious materials. Moreover, a thixotropic evaluation system has been established based on the proposed model to evaluate the thixotropic property of cement paste. This study is presented in Chapter 2. Then, we proposed a Navier-Stokes Informed Neural Network (NSINN) to study the flow behavior of cement paste in the 3DCP barrel and nozzle since it involves complicated Multiscale or Multiphysics behaviors, such as anisotropy and non-uniform shear rate distribution. The Navier Stokes Equation and the rheological constitutive equations are coupled and embedded into the NSINN architecture to simulate the flow behavior in the barrel during 3D-printing extrusion process. This approach demonstrates the potential of PINNs to provide high-accuracy simulations of velocity and pressure fields in a computationally efficient manner compared to traditional mesh-based models. Also, we used the NSINN to learn the relationship between the nozzle size and the printing quality. The details of the study are presented in Chapter 3. At last, a multi-subnetwork PINN architecture (MultiSubPINN) is proposed to address the limitations of traditional PINNs. This is because traditional PINNs used Fully-connected Neural Network (FNN) as main bone. It has problems like different outputs must share parameters within the network structure. However, this parameter sharing nature might lead to potential errors and decrease the model performance. The main architecture of this network is consisted of multiple separated sub-networks and the loss function of each sub-network is calculated separately to update the parameters in each sub-network. By doing this, we can ensure that the optimization for one output doesn't negatively impact others, therefore providing a more flexible, scalable, and potentially more accurate alternative for the evaluation. Besides, using randomly distributed training points in the computational domain for PINNs is a common way. However, many physical problems exhibit non-uniform behavior, with certain regions of the domain having more complex dynamics or sharper gradients than others. Randomly distributed training points might not adequately capture these regions, leading to a model that misrepresents the underlying physics in critical areas. We proposed a Physical-based sampling strategy to optimize the training of the PINNs. This study is presented in Chapter 4. In summary, this dissertation has established a new platform that is enabled by interpretable NN to accurately quantify cement paste’s rheological properties and simulate the flow behavior of 3D concrete printing process. The findings suggest that PINNs can not only improve the efficiency and accuracy of simulations but also provide a more profound understanding of the material behaviors and process dynamics critical to 3DCP. The novel PINN structure developed from this dissertation is a powerful general platform that could be applicable to a variety of material domains for complex rheological behavior predictions.

Partial differential equations (PDEs) are core tools for characterizing physical laws and are widely used in fluid mechanics, power systems, additive manufacturing, and other fields. However, traditional numerical methods are limited by mesh partitioning, leading to a trade-off between accuracy and efficiency in complex geometric domains and high-dimensional problems. While Physically Informed Neural Networks (PINNs) achieve meshless PDE solutions by embedding prior physical knowledge, they still face challenges such as insufficient accuracy, low training efficiency, and poor stability. This study aims to address the core bottlenecks of PINN in solving PDEs by proposing a systematic optimization strategy to improve its numerical solution accuracy, efficiency, and stability. Methodologically, firstly, a composite loss function containing PDE residuals and boundary initial condition constraints is constructed based on automatic differentiation techniques to ensure that the network satisfies physical laws. Secondly, the PINN training system is optimized, including using Neural Architecture Search (NAS-PINN) to automatically match the optimal network structure, designing an adaptive data sampling strategy to improve data utilization, and introducing parallel computing and hardware acceleration techniques to reduce training time. Finally, the effectiveness of the strategy is verified through classical PDEs and engineering problems. The results show that the optimized PINN achieves high-precision numerical approximation in classical PDE solutions and can efficiently solve forward design and backward parameter inversion problems in engineering scenarios. Compared with traditional methods, it exhibits stronger adaptability in complex geometric domains and high-dimensional problems, while improving training efficiency by more than 30% and significantly enhancing stability. This research not only enriches the optimization theory of PINN and provides an efficient new path for PDE numerical solutions, but also promotes the practical application of PINN in engineering fields such as fluid mechanics, additive manufacturing, and power systems, possessing significant theoretical value and application significance.

In recent engineering applications using deep learning, physics-informed neural network (PINN) is a new development as it can exploit the underlying physics of engineering systems. The novelty of PINN lies in the use of partial differential equations (PDE) for the loss function. Most PINNs are implemented using automatic differentiation (AD) for training the PDE loss functions. A lesser well-known study is the use of finite difference method (FDM) as an alternative. Unlike an AD based PINN, an immediate benefit of using a FDM based PINN is low implementation cost. In this paper, we propose the use of finite difference method for estimating the PDE loss functions in PINN. Our work is inspired by computational analysis in electromagnetic systems that traditionally solve Laplace’s equation using successive over-relaxation. In the case of Laplace’s equation, our PINN approach can be seen as taking the Laplacian filter response of the neural network output as the loss function. Thus, the implementation of PINN can be very simple. In our experiments, we tested PINN on Laplace’s equation and Burger’s equation. We showed that using FDM, PINN consistently outperforms non-PINN based deep learning. When comparing to AD based PINNs, we showed that our method is faster to compute as well as on par in terms of error reduction.

A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations

Satellite-based research on global sea surface wind is essential for understanding and monitoring the air-sea interface dynamical processes, driving the need for accurate and efficient assessment. Spaceborne scatterometers, which are among the most relevant sensors for sea surface wind observation, play a vital role in obtaining global ocean surface wind information. However, a significant challenge associated with polar-orbiting satellites lies in data gaps caused by their orbital paths, resulting in missing observations between swaths. While several satellite-derived sea surface wind products have been developed using data assimilation (DA) techniques, these existing methods are time-consuming and require large amounts of diverse data, rendering them computationally expensive. Additionally, the iterative steps of variational algorithms can only perform linear or weakly nonlinear adjustments to the governing equations, which may pose challenges given the highly non-linear nature of these equations.In this study, we leveraged physics-informed neural networks (PINNs) techniques to reconstruct large-scale sea surface wind fields with sparse scatterometer observations from different satellites, integrating observations and filling gaps. By incorporating physical constraints into the loss function, specifically the Navier-Stokes equations, we efficiently fill the data gaps and reconstruct wind fields that not only match observational data but also adhere to physical principles. Another objective of this work is to introduce the wind speed gradient and direction parallel consistent constraints into the loss function in order to enhance the detail of the reconstructed wind field and increase the accuracy of both wind speed and direction. Structurally, the PINN resembles a fully connected neural network (FCNN), offering the advantage of automatic feature extraction. Our model not only extracts valuable information from existing data but also uncovers complex patterns and correlations in data that are difficult for traditional algorithms to capture. This approach provides a novel perspective and an alternative methodology for wind field reconstruction.The results show that PINNs can reconstruct wind fields that closely resemble realistic wind patterns, capturing large-scale structures while preserving fine-scale details, thanks to the introduction of wind speed gradient and direction parallel consistent constraints. The training time for this model is about 3 hours, using only a single GPU core. This efficiency is partly due to the fact that PINNs do not rely on ensemble methods or large datasets to produce results. Unlike traditional DA methods, PINN does not depend on an initial best-guess field for assimilating observations. While we use only a small amount of scatterometer observation data, no initial field is required to complete the reconstruction. Additionally, since the PINN represents a continuous and differentiable function, it can produce outputs at any spatial or temporal point within the training domain.Recognizing their potential for forecast models and data integration, PINNs offer a promising approach for accurate sea surface wind field reconstruction and could serve as an effective alternative to current methods.

Numerical simulation and experimental techniques are the primary methods for solving fluid dynamics problems. However, while numerical simulation approaches are sensitive when meshing a complex structure, experimental methods have difficulty simulating the physical challenges. Therefore, building an affordable model to solve the fluid dynamics problem is very important. Deep learning (DL) approaches have great abilities to handle strong nonlinearity and high dimensionality that attract much attention for solving fluid dynamics problems. In this paper, we used a deep learning-based framework, physics-informed neural networks (PINNs). The main idea of PINN approaches is to encode the underlying physical law (i.e., the partial differential equation) into the neural network as prior information. In the oil recovery process involving the injection of fluids and multiphase flow in porous media, fingering instability is observed if a fluid with low viscosity displaces a high viscosity fluid. This paper provides a deep learning framework to simulate the instability (fingering) phenomenon during secondary and enhanced oil recovery methods. We have considered two models, namely, the first model on immiscible multiphase flow from the secondary oil recovery method, which is analyzed both with and without deliberating mass flow rate. In the second model, instability arising during the enhanced oil recovery method involving miscible displacement of multiphase is examined, focusing on oil recovery using carbonated water injection. We solve the governing nonlinear partial differential equation using PINNs. Furthermore, we have compared results from PINNs with the semi-analytical solution from the literature. The results show that PINNs are very effective in fluid flow problems and deserve further research.

In this paper, we propose an extrinsic approach based on physics-informed neural networks (PINNs) for solving the partial differential equations (PDEs) on surfaces embedded in high dimensional space. PINNs are one of the deep learning-based techniques. Based on the training data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. Using automatic differentiation, the PDEs information could be encoded into NNs and a loss function. To deal with the surface differential operators in the loss function, we combine the extrinsic approach with PINNs and then express that loss function in extrinsic form. Subsequently, the loss function could be minimized extrinsically with respect to the NN parameters. Numerical results demonstrate that the extrinsic approach based on PINNs for surface problems has good accuracy and higher efficiency compared with the embedding approach based on PINNs. In addition, the strong nonlinear mapping ability of NNs makes this approach robust in solving time-dependent nonlinear problems on more complex surfaces.

Summary Classical reservoir engineering studies require building geological models and solving complex fluid flow transport equations that require high-quality data, numerous computational resources, time and workflows. For large and mature fields data-driven models can be used to get faster answer and to perform production analysis more efficiently. Capacitance Resistive Models (CRM) are a class of methods based on material balance that can be used to estimate production wells liquid rates as a function of injected water and Bottom Hole Pressure (BHP) variations. CRM methods quantify the connectivity between producers and injectors using only dynamic data. An important drawback of CRM is that they can suffer from parameter identification problems. Moreover, the analytical solution can be only obtained in specific conditions: linear variations of BHP and fixed injection rate between two consecutive time steps. In this work we present a new approach combining CRM material balance equations with neural networks in order to obtain more robust and reliable estimation of the CRM parameters (i.e. well connectivity, productivity indices and time constants). This proposal is also interesting since it is not based on any assumption on BHP and injection rates. To this end, we use a recent approach called Physics Informed Neural Networks (PINNs). In this approach neural networks are trained on observed data with additional physics constraints traduced in appropriate loss functions. The parameters of this physical equation are evaluated at the same time as the neural network weights. The introduction of PINNs in our approach raised after testing classical machine learning (ML) models (SVMs, Random Forests …) and deep learning models (MLP, LSTM, RNNs…). Indeed, such models can perform well in some specific cases but usually struggle to produce robust results (i.e. forecasting) in the long term. Unfortunately, such systems do not natively integrate physics constraints. Our aim is to impose physic constraints in neural networks. Thus, we may obtain more stable and reliable results. On the same time, we should be able to account for more behaviors that are not explained by simplified physic equations such as material balance. We performed a full comparison between our approach using PINNs, other standard ML and DL approaches and a given framework of CRMs on two data-sets: a simple but realistic model build using a commercial reservoir simulator, and a real data set. We show that our approach gives more robust results (in terms of MSE) while not suffering from parameter identification issue.

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional partial differential equations (PDEs), as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerical PDEs in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs’ and PINNs’ residual into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton–Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in less than one hour for 1000 dimensions and in 12 h for 100,000 dimensions on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

A nonlocal physics-informed deep learning framework using the peridynamic differential operator

Identifying optimal architectures of physics-informed neural networks by evolutionary strategy

A physics-informed neural network (PINN) uses physics-augmented loss functions, e.g., incorporating the residual term from governing partial differential equations (PDEs), to ensure its output is consistent with fundamental physics laws. However, it turns out to be difficult to train an accurate PINN model for many problems in practice. In this article, we present a novel perspective of the merits of learning in sinusoidal spaces with PINNs. By analyzing behavior at model initialization, we first show that a PINN of increasing expressiveness induces an initial bias around <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">flat output functions</i> . Notably, this initial solution can be very close to satisfying many physics PDEs, i.e., falling into a local minimum of the PINN loss that only minimizes PDE residuals, while still being far from the true solution that jointly minimizes PDE residuals and the initial and/or boundary conditions. It is difficult for gradient descent optimization to escape from such a local minimum trap, often causing the training to stall. We then prove that the sinusoidal mapping of inputs—in an architecture we label as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sf</i> -PINN—is effective to increase input gradient variability, thus avoiding being trapped in such deceptive local minimum. The level of variability can be effectively modulated to match high-frequency patterns in the problem at hand. A key facet of this article is the comprehensive empirical study that demonstrates the efficacy of learning in sinusoidal spaces with PINNs for a wide range of forward and inverse modeling problems spanning multiple physics domains.

Physics-informed neural networks in water and wastewater systems: a critical review.

Error homogenization in physics-informed neural networks for modeling in manufacturing

Physics-informed neural networks (PINNs) have been widely used to solve various scientific computing problems. However, large training costs limit PINNs for some real-time applications. Although some works have been proposed to improve the training efficiency of PINNs, few consider the influence of initialization. To this end, we propose a New Reptile initialization based Physics-Informed Neural Network (NRPINN). The original Reptile algorithm is a meta-learning initialization method based on labeled data. PINNs can be trained with less labeled data or even without any labeled data by adding partial differential equations (PDEs) as a penalty term into the loss function. Inspired by this idea, we propose the new Reptile initialization to sample more tasks from the parameterized PDEs and adapt the penalty term of the loss. The new Reptile initialization can acquire initialization parameters from related tasks by supervised, unsupervised, and semi-supervised learning. Then, PINNs with initialization parameters can efficiently solve PDEs. Besides, the new Reptile initialization can also be used for the variants of PINNs. Finally, we demonstrate and verify the NRPINN considering both forward problems, including solving Poisson, Burgers, and Schr\"odinger equations, as well as inverse problems, where unknown parameters in the PDEs are estimated. Experimental results show that the NRPINN training is much faster and achieves higher accuracy than PINNs with other initialization methods.