Lately I found an interest in solving PDEs numerically. I've done some simulations on rectangular grids(heat equation, wave equation with location dependent velocity etc), and I'd like to do something more sophisticated. My end goal is to perform numerical simulations on curved surfaces, but simple things first.
Flat domain
Given I've got a flat domain discretized into a bunch of triangles with a value associated at each point. I would like to calculate the spatial derivatives at point P given the values at P's immediate neighbors. I figured I could calculate a linear function that interpolates values at vertices belonging to each neighboring triangle, calculate the derivatives based on each interpolant and then do a weighted average. However I'm not sure whether the weight should be with respect to the surface area S, or the edge length l?
Abstracting the ambient space
Is it possible/sensible to express the gradient/laplace operator not in terms of derivatives with respect to the axes of ambient space, but in terms of some sort local barycentric coordinates spanned on the neighboring vertices? Would the differential equation change in any way? E.g would the equation u_{t} = β^(2)u remain the same, or would some additional factors show up?
Curved domains
What considerations would I need to take if the domain weren't flat? For example if the domain would approximate the surface of a sphere or a torus? I feel like there's a need to account for the curvature of the surface in the differential operators. This is why I thought about expressing the gradient not with respect to ambient space but with respect to the local space defined by the simplicial complex(should it be a good idea).
All of this is for fun/learning, so one can assume everything is sufficiently nice, smooth and not an edge case.
(Forgive me if I didn't tag this correctly, wasn't sure what the best fit was).
For work, I want to implement the CMOS noise reduction algorithm that's in this paper:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7582543/
I'm having difficulty with this part, immediately preceding equation (8): https://imgur.com/2kQDjDH
- What does it mean to apply a Fourier transform operator to a differential operator that's by itself, as in the denominator? I would usually interpret a lone differential operator as being applied to an implied "1", but here that would just zero out those terms every time, and that doesn't seem right.
- Another question: in these lines, it appears that the matrix U is factored out from the left side of the equation. Does this imply that the identity matrix is left behind, and further that U must be square?
Thanks for any input.
Was wondering this these days, in particular in the context of getting more insight into the Laplace transform, by letting G(t,s) = exp(-st)
Croatians - Croatian linguistic purism i.e. making up words such as "zrakomlat", the infamous "mljevenik", etc. Making fun of "dakanje" e.g. acting as if it's grammatically incorrect to say "idem da radim" instead of "idem raditi".
Montenegrins - rejecting cyrillic, the addition of new letters such as "Ε" and "ΕΉ".
Bosniaks - Turkisms, throwing in the letter h, e.g., "lahko", "polahko", etc.
I'm curious, what have Serbs done to differentiate their standard? Who has kept their standard most similar to pre-war Serbo-Croatian? And whose changes are most acceptable, and whose most laughable?
I gave my earnest attempt at this question and could not get anywhere
Here is the question:https://ibb.co/nnHgsdd Some Context to the Question I leave there from Passages to the book https://ibb.co/album/BKQyx7
Here is the Attempt:https://ibb.co/album/R4NqKx Please can someone show me the details for this proof please?
View the full paper presentation here which includes a time-stamped outline:
Numerical solvers for Partial Differential Equations are notoriously slow. They need to evolve their state by tiny steps in order to stay accurate, and they need to repeat this for each new problem. Neural Fourier Operators, the architecture proposed in this paper, can evolve a PDE in time by a single forward pass, and do so for an entire family of PDEs, as long as the training set covers them well. By performing crucial operations only in Fourier Space, this new architecture is also independent of the discretization or sampling of the underlying signal and has the potential to speed up many scientific applications.
Abstract:
The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and the Navier-Stokes equation (including the turbulent regime). Our Fourier neural operator shows state-of-the-art performance compared to existing neural network methodologies and it is up to three orders of magnitude faster compared to traditional PDE solvers.
Authors: Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar
https://preview.redd.it/xi4s48hk11571.png?width=386&format=png&auto=webp&s=9b3799e9a4e033f33d170146a55bf10dcf19b37d
This new paper by researchers from CalTech & Purdue is notable for making significant advancements in solving Partial Differential Equations, critical for understanding the world around us. Here is a great video that explains the paper in detail. (You can click the time-stamps on the right to jump between sections.)
Quick Summary:
Numerical solvers for Partial Differential Equations are notoriously slow. They need to evolve their state by tiny steps in order to stay accurate, and they need to repeat this for each new problem. Neural Fourier Operators, the architecture proposed in this paper, can evolve a PDE in time by a single forward pass, and do so for an entire family of PDEs, as long as the training set covers them well. By performing crucial operations only in Fourier Space, this new architecture is also independent of the discretization or sampling of the underlying signal and has the potential to speed up many scientific applications.
More information about the research:
Paper: https://arxiv.org/abs/2010.08895
Code: https://github.com/zongyi-li/fourier_neural_operator/blob/master/fourier_3d.py
MIT Technology Review: https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/
The constant says, "sorry, e to the x, every time he comes in he kills me."
e to the x says, "huh, when I meet him nothing happens."
The constant leaves and the differential operator orders at the bar. e to the x decides to go over and say hi.
"Hi, I'm e to the x," e to the x says to the differential operator.
The differential operator responds, "hi, I'm d/dy."
I have a linear differential operator and its complementary solution u. Is there a way to show that the complementary solution of the adjoint operator is u/(p(x)*W) where p is the coefficient of the second order differential and W is the wronskian
i mean the curved d. some people just call it d, other call it doo/die, others call it partial. whatβs the right way to read it?
For the differential operator d^y/dx^2 with say dy/dx=0 for x=0 and x=1, we know that the Eigenvalues are \lambda_j=-(j\pi)^2 for j=0,1......
We know that the Eigenvalues of the operator d^2y/dx^2 +ay Are the same as above but shifted by a, when a is constant. However, can we estimate how much they shift when a=a(x)? Iβve looked in the literate and havenβt found an answer. This seems like a classical problem, so Iβm shocked that itβs been so hard to find an answer.
In my differential equations class we learned that when you have a second order homogeneous ODE and the differential operator is squared, using the substitution y=e^rx only produces 1 solutionβitβs not too difficult to find the other solution (xe^rx) in this case because the order is low, but when we have a higher order operator how do we show that multiplying the solution by x produces a new solution?
Hello r/math.
I'm reading in the book "Ordinary Differential Equations" by Tenenbaum and Pollard, and there's a chapter that introduces differential and polynomial operators.
They represent the derivative d/dx as D, and then you can have a polynomial in D that you can "multiply" with a function. D^2 y means d^2 y/dx^2, and (D+1)^2 y = D^2 y + 2Dy + y etc. The algebraic roots of the polynomial behave as expected in this new world. (They also represent Laplace transform with it, but I know Laplace well and I'm not looking for more on that.)
The method seems very powerful (like the Shift theorem), and it feels like there is a lot more to it than in that book. Especially on the algebra side of it. But I can't find any good references that are easy for me to understand.
From https://en.wikipedia.org/wiki/Differential_operator, I see that Weyl algebra may be the final destination. But that last page looks way too complicated to me.
Also I accidentally ran into a paper that justifies what used to be called "umbral calculus" with another form of "operators". So this stuff must be really good!
I'm comfortable with most undergrad topics, and know groups/rings/fields+ in abstract algebra, and happy to read anything as long as it's not 30 new advanced words for me in every paragraph :).
Do you have a good recommendation for a book/document that builds up that theory of "operators" well, but without assuming, like, advanced graduate stuff?
Thank you :).
I'm looking for some graduate texts on the subjects in the title and preferably some including relevant material on multilinear algebra and tensor analysis. I'm aware of do Carmo's Differential Geometry of Curves and Surfaces as well as Gravitation by Thorne et al. I'd like to supplant the former (whose language is a bit uncanonical) and be equipped to comprehend the latter. I did my undergrad in physics and math, have a good fundamental understanding of general topology, manifolds, and abstract algebra and have done work in special relativity, if that helps.
Happy labor day my fellow statisticians. As the title suggests, Iβm curious as to what the difference is, if any, between adding an autocorrelation term vs optimizing a dy/dt up to a specific order in a General Linear Regression ? Is one inherently biased or are they the same in terms of the Linear Algebra. Thanks for any insight!
I really tried for more than an hour.
I don't understand the part where: "This function has a jump", "This function has no jump but its derivative does", and "This function w(t) has no jump in value or derivative, but its second derivative does jump"
This new paper by researchers from CalTech & Purdue is notable for making significant advancements in solving Partial Differential Equations, critical for understanding the world around us. Here is a great video that explains the paper in detail. (You can click the time-stamps on the right to jump between sections.)
Quick Summary:
Numerical solvers for Partial Differential Equations are notoriously slow. They need to evolve their state by tiny steps in order to stay accurate, and they need to repeat this for each new problem. Neural Fourier Operators, the architecture proposed in this paper, can evolve a PDE in time by a single forward pass, and do so for an entire family of PDEs, as long as the training set covers them well. By performing crucial operations only in Fourier Space, this new architecture is also independent of the discretization or sampling of the underlying signal and has the potential to speed up many scientific applications.
More information about the research:
Paper: https://arxiv.org/abs/2010.08895
Code: https://github.com/zongyi-li/fourier_neural_operator/blob/master/fourier_3d.py
MIT Technology Review: https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/
Numerical solvers for Partial Differential Equations are notoriously slow. They need to evolve their state by tiny steps in order to stay accurate, and they need to repeat this for each new problem. Neural Fourier Operators, the architecture proposed in this paper, can evolve a PDE in time by a single forward pass, and do so for an entire family of PDEs, as long as the training set covers them well. By performing crucial operations only in Fourier Space, this new architecture is also independent of the discretization or sampling of the underlying signal and has the potential to speed up many scientific applications.
OUTLINE:
0:00 - Intro & Overview
6:15 - Navier Stokes Problem Statement
11:00 - Formal Problem Definition
15:00 - Neural Operator
31:30 - Fourier Neural Operator
48:15 - Experimental Examples
50:35 - Code Walkthrough
1:01:00 - Summary & Conclusion
Paper: https://arxiv.org/abs/2010.08895
Blog: https://zongyi-li.github.io/blog/2020/fourier-pde/
Code: https://github.com/zongyi-li/fourier_neural_operator/blob/master/fourier_3d.py
MIT Technology Review: https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/
