Bogdan Raiță
Department of Mathematics and Statistics
Georgetown University
With many, many contributions by Marc Härkönen, Markus Lange-Hegermann, Jianlei Huang, and Xin Li
\[\frac{\mathrm{d}^2y}{\mathrm{d}t^2} + 2 \frac{\mathrm{d}y}{\mathrm{d}t} + 10y = 0\] 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 $R=\R[\partial]$ where $\partial=(\partial_{x_1},\ldots,\partial_{x_n})$.
Want to solve $A(\partial)f=0$ for $f=(f_1,f_2,\ldots,f_\ell)\in \mathcal F^\ell$,
where $\mathcal F$ is a space of functions,
e.g. $\mathcal F=C^\infty(\Omega)$.
Thus $\mathcal F$ is a left $R$-module under the action of differentiation.
All solutions are linear combination of exponential-polynomial solutions up to approximation.
Let $A \in R^{\ell' \times {\ell}}$ for $R=\R[\partial]$ where $\partial=(\partial_{x_1},\ldots,\partial_{x_n})$.
Want to solve $Af=0$ for $f=(f_1,f_2,\ldots,f_\ell)\in \mathcal F^\ell$, where $\mathcal F$ is a space of functions, e.g. $\mathcal F=C^\infty(\Omega)$.
Thus $\mathcal F$ is a left $R$-module under the action of differentiation.
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 $R=\R[\partial]$ where $\partial=(\partial_{x_1},\ldots,\partial_{x_n})$.
Want to solve $A(\partial)f=0$ for $f=(f_1,f_2,\ldots,f_\ell)\in \mathcal F^\ell$, where $\mathcal F$ is a space of functions, e.g. $\mathcal F=C^\infty(\Omega)$.
Thus $\mathcal F$ is a left $R$-module under the action of differentiation.
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 $R=\R[\partial]$ where $\partial=(\partial_{x_1},\ldots,\partial_{x_n})$.
Want to solve $A(\partial)f=0$ for $f=(f_1,f_2,\ldots,f_\ell)\in \mathcal F^\ell$, where $\mathcal F$ is a space of functions, e.g. $\mathcal F=C^\infty(\Omega)$.
Thus $\mathcal F$ is a left $R$-module under the action of differentiation.
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 $R=\mathbb{C}[\partial_{x_1},\ldots,\partial_{x_n}]$, $\Omega \subseteq \R^n$ be convex open, and $\mathcal F=C^\infty(\Omega)$. Then \[ \begin{align*} \ker_{\mathcal F}A=\{f\in \mathcal F^\ell\colon A(\partial) f=0 \}\simeq \dfrac{R^\ell}{\mathrm{im}_{R}A^\top}. \end{align*} \] Primary decomposition: $\mathrm{im}_R A^\top=\bigcap_{i=1}^s \mathrm{im}_R A_i^\top$ yields $\ker_{\mathcal F}A=\sum_{i=1}^s \ker_{\mathcal F}A_i$. There exist
Recent work: Cid-Ruiz-Homs-Sturmfels '21, Chen-Cid-Ruiz-Härkönen-Krone-Leykin '21.
Old work: Palamodov, Ehrenpreis '70, Hörmander '73, Björk '79, Eisenbud-Huneke-Vasconcelos '92.
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, open set. There exist
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, open 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
\[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \]