Equation Solver Geometry
Leanne Wittlin
This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of artificial intelligence large language models to parse and generate natural language answers. This creates a math problem solver that's more accurate than ChatGPT, more flexible than a math calculator, and provides answers faster than a human tutor.
Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects.
Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.
Surrogate models, which approximate the output in relation to the input, could increase the use in these fields. Quantities like the fluid velocity and pressure could be captured very fast by these methods. Three different types of surrogate modeling techniques can be distinguished: Reduced Order Models (ROM), data fit models and Deep Neural Network (DNN) based models.
Data fit models create a fit between input and output based on simulations. The most popular methods are polynomial basis, radial basis functions, Gaussian process and stochastic polynomial chaos expansion. These methods mostly do not require any change in the simulation solver.
Our current work focuses on advancing the PDE-constrained deep learning framework for more real-world applications with irregular geometric shapes. The paper is organized as follows. First the physical background and its mathematical description is presented. In the next section the architecture of the deep learning algorithm proposed here is explained. In the following chapters the numerical experiments will be described and are followed by the results obtained from the proposed GAPINN framework in comparence with CFD-Simulations and vanilla PINN. Finally, conclusions are drawn in the last chapter.
The GAPINN framework consists of three separate networks, see Fig. 1: (1) as one of the most important parts, to solve for varying non-parametric geometries, a Shape Encoding Network (SEN); (2) a Physics Informed Neural Network (PINN) in order to solve the differential equation of a given fluid mechanical problem; (3) and a Boundary Constrain Network (BCN) to constrain the boundary and initial conditions for each given non-parametric geometric boundaries.
Schematic description of the architecture of the proposed DNN Method (GAPINN) to generate surrogates of PDEs with irregular non-parameterized geometries using PINNs. The network consists of three subnetworks which are trained separately. The SEN is a Variational-Auto-Encoder type reducing the geometry dimensions to a latent vector k. PINN takes k and spatial positions to solve the PDE and building the surrogate and BCN, by also taking spatial information and k, helps constraining boundary conditions in the PINN. Dimensions at each operation are noted in brackets
We first describe the SEN more in detail to help the reader to understand how different fluid domain geometries can be interpreted by a PINN. Moreover, how this facilitates the development of a surrogate model that is able to solve fluid mechanical partial differential equations for various non-parametric geometries without the need of training data.
As input we assumed a non-parametric but well-defined fluid domain. We aimed to get a latent representation of each geometric shape, by using the technique of Variational-Auto-Encoder (VAE) [9]. VAE are a common technique in the field of computer vision to reduce high dimensional information to a lower dimensional representation in an unsupervised learning process. A VAE is built from two main components namely the encoder, which actually reduces the dimensions and the decoder, which reconstructs the input from a lower dimensional representation.
A reason why we recommend VAE instead of Auto-Encoder (AE) is that we had found poor validation performance for AE. In order to obtain a feasible latent representation, in terms of interpolation capabilities for geometric similar shapes, we used a VAE with a regularization term which fits the latent vector to a known distribution.
For this loss function several first and second order derivatives of the output (U,p) with respect to spatial coordinates X were needed. These calculations were performed be means of the concept of automatic differentiation (AD). AD is a common technique used in the field of machine-learning, mainly to get gradients of the network with respect to the weights and biases. This technique relies on the concept of the calculation of derivatives inside a computational graph and is implemented in most state of the art deep-learning libraries such as TensorFlow, PyTorch or Theano; here we worked with PyTorch. For solving the optimization problem (Eq. 5), we used the Adam algorithm [16].
Boundary conditions can be imposed mainly in two ways. First, by adding an extra penalty loss term \(\mathcalL_soft\) to Eq. 4, which affects the PINN to learn conditions on the boundary, by minimizing \(\mathcalL\left( \mathbfW,\mathbfb \right)\) during training, with W and b being weights and biases of the neural network see Eq. 6. Sun et al. showed several major drawbacks of this so-called soft boundary imposing. As mentioned by Sun et al. this approach is not feasible to ensure the accurate definition of initial and boundary conditions due to its implicit manner. Furthermore, the optimization performance could depend on relative importance of the boundary loss term and PDE loss term [2]. This could be addressed by weighting the terms, but also a-prior weighting will mostly not be known.
The network takes as input the spatial positions \(\mathbfX_\textj,\left( \texti,\textb \right)\) and the latent vector k. The prediction of BCN was compared to pre-computed Euclidean distances of the spatial positions to the boundaries with fixed zero velocity by using mean squared error. Reduction of the mean squared error was done in the training process of the BCN, by adapting the weights of the neural network. The exact form of the functions B and C are highly dependent on the investigated fluid mechanical problem to be solved. For detailed description on how to construct the boundary functions for a specific problem we refer to the following experiment chapter.
Schematic of irregular geometric vessel with a fixed set of boundary and fluid domain points. With Dirichlet and Neumann boundary conditions: parabolic velocity profile at the inlet, zero velocity at walls and zero pressure at the outlet as well as zero gradient in reference to normal vectors (n) of geometry for pressure on inlet and vessel walls and velocity at the outlet. Red points are indicating internal points and green points are indicating points on the boundary
To validate the prediction performance of the data-free trained GAPINN we performed numerical simulations on randomly generated vessels by means computational fluid dynamics (CFD) using OpenFOAM 9 [18] for comparison with geometries used in training and ANSYS 18.0 Fluent with vessel geometries not included in training. The mesh consisted of 150,000 hexahedron elements. This validation set of vessels was not included in the training process.
To show the advantage of the developed framework, we also performed experiments on exactly the same dataset and training parameters while using Vanilla PINN framework. The Vanilla PINN had the same number of hidden layers and neurons in the hidden layers as the PINN used in the GAPINN approach. Furthermore, in addition to the hard boundary constraint strategy as presented here, we applied soft constraint strategy for both GAPINN and classical PINN.
Here we present results to evaluate the performance of the proposed DNN framework named GAPINN. We performed experiments on 2D steady and laminar flow inside vessels with irregular geometries referring to biomedical blood flow problem. We trained the three networks (SEN, BCN, PINN) on 1000 different geometries, referred to as training set. For evaluation of GAPINN we generated vessel that are not included in training, referred to as validation set. The output from the GAPINN were velocity u and v as well as pressure p. For comparing, the output data of GAPINN and CFD were interpolated on the same grid using linear interpolation.
Figure 4 shows the comparison between GAPINN predictions and CFD solutions on samples from the training and from the validation set with velocity in stream-wise component u on the left and span-wise velocity component v on the right.
The advantage of the GAPINN framework is that the quantities of interest can be estimated at any arbitrary point within the domain. Only the number of points describing the geometry should sufficiently represent the geometry and must be evaluated based on the given problem. But in general, the network architecture of all networks shown here is not limited in the number of points to be evaluated, in contrast to [8]. A disadvantage using point based computation compared to methods based on structured spatial representation is the higher computational effort to calculate the derivatives by means of AD, during training of the PINNs. Gao et al. showed this advantage by the usage of Finite Difference methods implemented in efficient kernel operations. The chosen encoder structure provides a permutation invariant processing of the geometry describing points [10]. Depending on the problem, other architectures are also possible as encoders. For example, it is reasonable to use convolution operations for geometries that do not contain permutation problems, e.g. if a unique sorting of the spatial information can be done. It would also be conceivable to obtain the latent representation by 2D or 3D CNN on the basis of image data representing the geometry, which could be feasible for example in imaging based medical examinations. Apart from reduction of geometric dimension using VAE, it could be possible to use other techniques of dimension reduction such as PCA.
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