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# System prepended metadata

title: Current Progress and Questions on Lie Symmetries

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# Current Progress and Questions on Lie Symmetries

### 1. On Robust Learning Paper:

**The general idea of a PINN:** to solve a PDE  with regard to a set of finite-order differential operators acting on the scalar fields and the source function (an analytical expression to solve a forward PDE), subject to a set of boundary conditions, to find the spatial-temporal evolution of the set of scalar fields by a deep neural network.

**The deep network:** calculates its parameters by minimizing the loss function, which is the composition of the mean squared errors of boundary samples and collocation points time their respective penalty parameters.

**The general idea of a cPINN:** training a PINN using discretized grid of evaluation points. (advantage: flux continuity, preventing error propagation.)

**Error propagation**: Overfitting on the boundaries at the initial timeslice, and the errors would propagate even with regularizers including functional and flux continuity at interfaces and conservation constraints.

**Gaussian process based error correction**: A smoothing technique by modeling the field in the initial timeslice by a pair of continuous stochastic processes and then obtaining a funcitonal estimate of them by using Gaussian Process Regression, whose kernal function the radial basis function appended with a white noise function.

* RBF: the kernel usually used in support vector machine classifications
* Matèrn: generalized Gaussian RBF
* Rational Quadratic: an infinite sum of different exponentiated quadratic kernels

** **


### 2. On Lie Symmetries:

**Symmetry:** an operation in which a physical property has an equivalence in two or more directions. (Object --Transformation--> Identical Object)
* translation
* rotation
* reflection
    
**Noether's Theorem:** Every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law.
    (Note: Conserved physical quantities = generators)
    
Lie groups correspond to the symmetries of smooth surfaces, including the Lagrangian surface.

**infinitesimal generator**


### 3. On Determining Equations Paper:


## Questions
1. On page 3, why is $u$<sub>D</sub> not necessary for training a PINN?

2. On equation 13, are u and v orthogonal functions? And why can we split the left side of equation 12 into two zero terms, instead of two opposite terms?

3. A lot of concepts in this paper are new to me, so I spent some time understanding things like Gaussian Process Regression, but to understand how to choose the number of hidden layers and nodes in an MLP, I need some help pointing to resources.

4. Why are the predefined Gaussian errors sigmas different for Schrodinger equation and Burgers' equation, also their respective timeslices.

