Markus Lange-Hegermann
Institut für industrielle Informationstechnik - inIT
Department of Electrical Engineering and Computer Science, TH OWL
Bogdan Raiță
Department of Mathematics and Statistics
Georgetown University
With many, many contributions by Andreas Besginow, Marc Härkönen, Jianlei Huang, Xin Li, and Daniel Robertz
\[\frac{\mathrm{d}^2y}{\mathrm{d}t^2} + 2 \frac{\mathrm{d}y}{\mathrm{d}t} + 10y = 0\] Sample 5 noisy points
How to solve? \[\frac{\mathrm{d}^2y}{\mathrm{d}t^2} + 2 \frac{\mathrm{d}y}{\mathrm{d}t} + 10y = 0\]
Algebraic preprocessing: characteristic frequencies \[z^2 + 2z + 10 = 0 \Rightarrow z = -1 \pm 3\sqrt{-1}\]
Solution Space \[y(t) = (\textcolor{#b51963}{c_1} \cdot e^{-t} \cos 3t + \textcolor{#b51963}{c_2} \cdot e^{-t} \sin 3t)\]
How to solve? \[\frac{\mathrm{d}^2y}{\mathrm{d}t^2} + 2 \frac{\mathrm{d}y}{\mathrm{d}t} + 10y = 0\]
Algebraic preprocessing: characteristic frequencies \[z^2 + 2z + 10 = 0 \Rightarrow z = -1 \pm 3\sqrt{-1}\]
Solution Space \[y(t) = (\textcolor{#b51963}{c_1} \cdot e^{-t} \cos 3t + \textcolor{#b51963}{c_2} \cdot e^{-t} \sin 3t)\]
Gaussian process prior \[Y(t) \sim (\textcolor{#b51963}{C_1} \cdot e^{-t} \cos 3t + \textcolor{#b51963}{C_2} \cdot e^{-t} \sin 3t) + \textcolor{#b51963}{\epsilon}\] where \[ \textcolor{#b51963}{C_1 \sim \mathcal{N}(0, \sigma_1)}, \textcolor{#b51963}{C_2 \sim \mathcal{N}(0, \sigma_2)}, \textcolor{#b51963}{\epsilon \sim \mathcal{N}(0, \sigma_0)} \]
How to solve? \[\frac{\mathrm{d}^2y}{\mathrm{d}t^2} + 2 \frac{\mathrm{d}y}{\mathrm{d}t} + 10y = 0\]
Algebraic preprocessing: characteristic frequencies \[z^2 + 2z + 10 = 0 \Rightarrow z = -1 \pm 3\sqrt{-1}\]
Solution Space \[y(t) = (\textcolor{#b51963}{c_1} \cdot e^{-t} \cos 3t + \textcolor{#b51963}{c_2} \cdot e^{-t} \sin 3t)\]
Gaussian process prior \[Y(t) \sim (\textcolor{#b51963}{C_1} \cdot e^{-t} \cos 3t + \textcolor{#b51963}{C_2} \cdot e^{-t} \sin 3t) + \textcolor{#b51963}{\epsilon}\] where \[ \textcolor{#b51963}{C_1 \sim \mathcal{N}(0, \sigma_1)}, \textcolor{#b51963}{C_2 \sim \mathcal{N}(0, \sigma_2)}, \textcolor{#b51963}{\epsilon \sim \mathcal{N}(0, \sigma_0)} \]
Algebra: determine suitable frequencies
Stochastic: weigh frequencies
EPGP (ours) | S-EPGP(ours) | Vanilla PINN | Vanilla Num. Solver | |
---|---|---|---|---|
Differential equations | linear, c.c. | linear, c.c. | any | well-posed |
Additional Information | data | data | data | IC/BC |
Evaluation time | $\approx$ data | $\approx$ sparsity | neural net | mediocre |
Solutions? | yes | yes | approx. near data | yes |
Prerequisites | integral | - | - | - |
Sampling | yes | yes | no | not applicable |
How to solve? \[\frac{\mathrm{d}^4y}{\mathrm{d}t^4} + 2 \frac{\mathrm{d}^2y}{\mathrm{d}t^2} + y = 0\]
How to solve? \[\frac{\mathrm{d}^4y}{\mathrm{d}t^4} + 2 \frac{\mathrm{d}^2y}{\mathrm{d}t^2} + y = 0\]
Algebraic preprocessing: characteristic frequencies \[z^4 + 2z^2 + 1 = 0 \Rightarrow z = \pm i\text{ with multiplicity 2}\]
How to solve? \[\frac{\mathrm{d}^4y}{\mathrm{d}t^4} + 2 \frac{\mathrm{d}^2y}{\mathrm{d}t^2} + y = 0\]
Algebraic preprocessing: characteristic frequencies \[z^4 + 2z^2 + 1 = 0 \Rightarrow z = \pm i\text{ with multiplicity 2}\]
Solution Space \[y(t) = \textcolor{#b51963}{c_1}\textcolor{#0073e6}{t}e^{\textcolor{#0073e6}{-i}t} + \textcolor{#b51963}{c_2}e^{\textcolor{#0073e6}{-i}t}+\textcolor{#b51963}{c_3}\textcolor{#0073e6}{t}e^{\textcolor{#0073e6}{i}t}+\textcolor{#b51963}{c_4}e^{\textcolor{#0073e6}{i}t}\]
How to solve? \[\frac{\mathrm{d}^4y}{\mathrm{d}t^4} + 2 \frac{\mathrm{d}^2y}{\mathrm{d}t^2} + y = 0\]
Algebraic preprocessing: characteristic frequencies \[z^4 + 2z^2 + 1 = 0 \Rightarrow z = \pm i\text{ with multiplicity 2}\]
Solution Space \[y(t) = \textcolor{#b51963}{c_1}\textcolor{#0073e6}{t}e^{\textcolor{#0073e6}{-i}t} + \textcolor{#b51963}{c_2}e^{\textcolor{#0073e6}{-i}t}+\textcolor{#b51963}{c_3}\textcolor{#0073e6}{t}e^{\textcolor{#0073e6}{i}t}+\textcolor{#b51963}{c_4}e^{\textcolor{#0073e6}{i}t}\]
Gaussian process prior \[Y(t) \sim \textcolor{#b51963}{C_1}\textcolor{#0073e6}{t}e^{\textcolor{#0073e6}{-i}t} + \textcolor{#b51963}{C_2}e^{\textcolor{#0073e6}{-i}t}+\textcolor{#b51963}{C_3}\textcolor{#0073e6}{t}e^{\textcolor{#0073e6}{i}t}+\textcolor{#b51963}{C_4}e^{\textcolor{#0073e6}{i}t} + \textcolor{#b51963}{\epsilon}\] where \[ \textcolor{#b51963}{C_j \sim \mathcal{N}(0, \sigma_j)}, \textcolor{#b51963}{\epsilon \sim \mathcal{N}(0, \sigma_0)} \]
How to solve? \[\sum_{j=0}^k \alpha_j\frac{\mathrm d^jy}{\mathrm d t^j} = 0\]
How to solve? \[\sum_{j=0}^k \alpha_j\frac{\mathrm d^jy}{\mathrm d t^j} = 0\]
Algebraic preprocessing: characteristic frequencies \[\sum_{j=0}^k \alpha_jz^j=0 \Rightarrow z\in V = \text{ complex roots with multiplicities}\]
How to solve? \[\sum_{j=0}^k \alpha_j\frac{\mathrm d^jy}{\mathrm d t^j} = 0\]
Algebraic preprocessing: characteristic frequencies \[\sum_{j=0}^k \alpha_jz^k=0 \Rightarrow z\in V = \text{ complex roots with multiplicities}\]
Solution Space \[y(t) = \sum_{\textcolor{#0073e6}{z}\in V}\sum_{\text{multipliers for z}}\textcolor{#b51963}{c_{h}}\textcolor{#0073e6}{B_h(t)}e^{\textcolor{#0073e6}{z}t} \]
How to solve? \[\sum_{j=0}^k \alpha_j\frac{\mathrm d^jy}{\mathrm d t^j} = 0\]
Algebraic preprocessing: characteristic frequencies \[\sum_{j=0}^k \alpha_jz^k=0 \Rightarrow z\in V = \text{ complex roots with multiplicities}\]
Solution Space \[y(t) = \sum_{\textcolor{#0073e6}{z}\in V}\sum_{\text{multipliers for z}}\textcolor{#b51963}{c_{h}}\textcolor{#0073e6}{B_h(t)}e^{\textcolor{#0073e6}{z}t} \]
Principle for ODEs:
All solutions are linear combination of exponential-polynomial solutions.
\[\frac{\partial T}{\partial t}-\frac{\partial^2 T}{\partial x^2}=0\]
\[\frac{\partial T}{\partial t}-\frac{\partial^2 T}{\partial x^2}=0\]
All solutions are linear combination of exponential-polynomial solutions up to approximation.
Let $A \in R^{\ell' \times {\ell}}$ for $R=\R[\partial_{x_1},\ldots,\partial_{x_n}]$
Want to solve $Af=0$ for $f=(f_1,f_2,\ldots,f_\ell)$.
All solutions are linear combination of exponential-polynomial solutions up to approximation.
Let $A \in R^{\ell' \times {\ell}}$ for $R=\R[\partial_{x_1},\ldots,\partial_{x_n}]$
Want to solve $Af=0$ for $f=(f_1,f_2,\ldots,f_\ell)$.
Algebraic preprocessing: characteristic frequencies \[\ker A(z)=\{0\} \Rightarrow z\in V = \text{ characteristic variety}\]
All solutions are linear combination of exponential-polynomial solutions up to approximation.
Let $A \in R^{\ell' \times {\ell}}$ for $R=\R[\partial_{x_1},\ldots,\partial_{x_n}]$
Want to solve $Af=0$ for $f=(f_1,f_2,\ldots,f_\ell)$.
Algebraic preprocessing: characteristic frequencies \[\ker A(z)=\{0\} \Rightarrow z\in V = \text{ characteristic variety}\]
Solution Space is the closure of \[f(x) = \sum_{j=1}^M\sum_{h=1}^m\textcolor{#b51963}{c_{j}}\textcolor{#0073e6}{B_h(x,z_j)}e^{\textcolor{#0073e6}{z_j}\cdot x} \] where $\textcolor{#0073e6}{z_j}\in V$.
All solutions are linear combination of exponential-polynomial solutions up to approximation.
Let $A \in R^{\ell' \times {\ell}}$ for $R=\R[\partial_{x_1},\ldots,\partial_{x_n}]$
Want to solve $Af=0$ for $f=(f_1,f_2,\ldots,f_\ell)$.
Algebraic preprocessing: characteristic frequencies \[\ker A(z)=\{0\} \Rightarrow z\in V = \text{ characteristic variety}\]
Solution Space is the closure of \[f(x) = \sum_{j=1}^M\sum_{h=1}^m\textcolor{#b51963}{c_{j}}\textcolor{#0073e6}{B_h(x,z_j)}e^{\textcolor{#0073e6}{z_j}\cdot x} \] where $\textcolor{#0073e6}{z_j}\in V$.
This is the FUNDAMENTAL PRINCIPLE of Ehrenpreis-Palamodov ('70).
Explain Ehrenpreis-Palamodov in more detail:
Let $A \in R^{\ell' \times {\ell}}$ for $R=\R[\partial_{x_1},\ldots,\partial_{x_n}]$ and let $\Omega \subseteq \R^n$ be a convex, compact set. There exist
Sparse version of EPGP.
Sparse version of EPGP. Suitable, when integration is not possible.
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
Numerical solution:\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[\mathbf{E}(x,y,z,t)\]
\[ \mathbf{B}(x,y,z,t) \]
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]
\[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \]
\[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \]
Literature
\[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \]