Usando Inteligência Artificial para determinação dos graus de polinômios de Darboux de equações racionais
André Claudino
Método de Darboux
Dado uma 1DO racional
dxdy=N(x,y)M(x,y)
\dfrac{dy}{dx} = \dfrac{M(x,y)}{N(x,y)}
Existe um fator integrante...
R=∏ipi(x,y)ni
R = \prod_i p_i(x,y)^{n_i}
ni∈Q
n_i \in \mathbb{Q}
Que satisfaz...
RD[R]=−(Nx+My)
\dfrac{\mathbb{D}[R]}{R} = -(N_x + M_y)
∂y∂(RM)=−∂x∂(RN)
\dfrac{\partial (RM)}{\partial y} = - \dfrac{\partial (RN)}{\partial x}
onde...
D≡N(x,y)∂x+M(x,y)∂y
\mathbb{D} \equiv N(x,y) \partial_x + M(x,y) \partial_y
RD[R]=−(Nx+My)
\dfrac{\mathbb{D}[R]}{R} = -(N_x + M_y)
R=∏ipi(x,y)ni
R = \prod_i p_i(x,y)^{n_i}
⇒
\Rightarrow
∑inipiD[pi]=−(Nx+My)
\sum_i n_i \dfrac{\mathbb{D}[p_i]}{p_i} = -(N_x + M_y)
D[pi(x,y)]=pi(x,y)q(x,y)
\mathbb{D}[p_i(x,y)] = p_i(x,y) q(x,y)
∑inipiD[pi]=−(Nx+My)
\sum_i n_i \dfrac{\mathbb{D}[p_i]}{p_i} = -(N_x + M_y)
⇒
\Rightarrow
D[pi(x,y)]=pi(x,y)q(x,y)
\mathbb{D}[p_i(x,y)] = p_i(x,y) q(x,y)
O problema se torna resolver essa equação
Solução
Avellar, J. & Claudino, A.L.G.C. & Duarte, L. & Da Mota, Luis. (2015). Finding higher order Darboux polynomials for a family of rational first order ordinary differential equations. Computer Physics Communications. 195. 10.1016/j.cpc.2015.04.026.
Graus Máximos
Introdução
- Não existe limite superior teórico para os graus dos polinômios de Darboux
- Todos os métodos são semi-algorítmicos
Objetivo
Estimar os graus em que existam polinômios de Darboux
Semântica
- Representação vetorial de objetos com métrica que faça sentido para comparações no problema, baseada nas relaçõe
Frases e Palavras
- m1: Rock and Roll Music in the 1960’s
- m2: Recipe for a Drum Roll, a Demonstration of Techniques
- m3: Drum and Bass Composition
- m4: A Perspective of Rock Music in the 90’s
- m5: Music and Composition of Popular Bands
- b1: How to make Bread and Rolls, a Demonstration
- b2: Ingredients for Crescent Roll Dough
- b3: A Recipe for Sourdough Bread
- b4: A Quick Recipe for Pizza Dough with Natural Ingredients
Frases e Palavras
- m1: Rock and Roll Music in the 1960’s
- m2: Recipe for a Drum Roll, a Demonstration of Techniques
- m3: Drum and Bass Composition
- m4: A Perspective of Rock Music in the 90’s
- m5: Music and Composition of Popular Bands
- b1: How to make Bread and Rolls, a Demonstration
- b2: Ingredients for Crescent Roll Dough
- b3: A Recipe for Sourdough Bread
- b4: A Quick Recipe for Pizza Dough with Natural Ingredients
L=0.01.00.00.00.00.00.01.01.01.00.00.01.00.00.00.00.00.00.01.01.00.00.00.00.00.00.00.00.00.00.00.01.00.01.00.01.00.00.01.01.00.00.00.00.00.00.00.00.00.00.01.01.01.00.01.00.00.00.01.00.00.00.00.00.00.00.00.00.01.00.01.00.00.01.00.01.00.00.00.00.0
L = \left[\begin{matrix}0.0 & 1.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 0.0\\1.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0\\0.0 & 1.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 1.0\\0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 1.0 & 0.0\\1.0 & 0.0 & 0.0 & 1.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0\\1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 1.0 & 0.0\end{matrix}\right]
L=UWVT
L = UWV^T
Análise de componentes principais...
V=
V =
U=
U =
W=2.60.00.00.00.00.00.00.00.00.02.190.00.00.00.00.00.00.00.00.01.910.00.00.00.00.00.00.00.00.01.590.00.00.00.00.00.00.00.00.01.260.00.00.00.00.00.00.00.00.00.990.00.00.00.00.00.00.00.00.00.720.00.00.00.00.00.00.00.00.00.240.00.00.00.00.00.00.00.00.00.0
W = \displaystyle \left[\begin{matrix}2.6 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 2.19 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 1.91 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 1.59 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 1.26 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.99 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.72 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.24 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\end{matrix}\right]
Vetores semânticos
V3=−0.590.0−0.17−0.29−0.01−0.58−0.22−0.36−0.040.0−0.54−0.120.0−0.760.05−0.030.04−0.31−0.150.16−0.54−0.170.140.38−0.510.36−0.24
V_3 = \displaystyle \left[\begin{matrix}-0.59 & 0.0 & -0.15\\0.0 & -0.54 & 0.16\\-0.17 & -0.12 & -0.54\\-0.29 & 0.0 & -0.17\\-0.01 & -0.76 & 0.14\\-0.58 & 0.05 & 0.38\\-0.22 & -0.03 & -0.51\\-0.36 & 0.04 & 0.36\\-0.04 & -0.31 & -0.24\end{matrix}\right]
Vetores semânticos
U4=0.0−0.38−0.080.0−0.02−0.19−0.36−0.56−0.59−0.59−0.07−0.19−0.59−0.49−0.010.040.030.040.16−0.63−0.40.16−0.05−0.350.390.030.31−0.210.00.59−0.210.31−0.520.23−0.280.2
U_4 = \displaystyle \left[\begin{matrix}0.0 & -0.59 & 0.16 & -0.21\\-0.38 & -0.07 & -0.63 & 0.0\\-0.08 & -0.19 & -0.4 & 0.59\\0.0 & -0.59 & 0.16 & -0.21\\-0.02 & -0.49 & -0.05 & 0.31\\-0.19 & -0.01 & -0.35 & -0.52\\-0.36 & 0.04 & 0.39 & 0.23\\-0.56 & 0.03 & 0.03 & -0.28\\-0.59 & 0.04 & 0.31 & 0.2\end{matrix}\right]
Semântica de polinômios
p1=a1x2y1+a2x2y2
p_1 = a_1 x^2 y^1 + a_2 x^2 y^2
p2=b1xy1+b2x2y2
p_2 = b_1 x y^1 + b_2 x^2 y^2
⇒
\Rightarrow
0xy1+a1x2y1+a2x2y2
0 x y^1 + a_1 x^2 y^1 + a_2 x^2 y^2
b1xy1+0x2y1+b2x2y2
b_1 x y^1 + 0 x^2 y^1 + b_2 x^2 y^2
(0b1a10a2a2)
\begin{pmatrix}
0 & a_1 & a_2 \\
b_1 & 0 & a_2
\end{pmatrix}
xy1
x y^1
x2y1
x^2 y^1
x2y2
x^2 y^2
p1
p_1
p2
p_2
Preparação dos dados
- Separar numeradores e denominadores das equações
- Cria semânticas separadas para numeradores e denominadores
- Normaliza os vetores
Modelo neural
- Entradas numerador e denominador
- Reúne ambos no bloco racional
- Codificação com redução
- Prediz grau
Accurácia em torno de 80%
Projetos futuros
- Gerar métrica por triplet loss
- Modelo de contraste
- Aprendizado one-shot para graus novos
Perguntas?
Usando Inteligência Artificial para determinação dos graus de polinômios de Darboux de equações racionais André Claudino