Scientific Machine learning Using Julia

This repoitory showcases the solution to Ordinary and Partial Differential Equations.

Ordinary Differential Equations

Oscilation of a pendulum

Equations:

dθ(t)/dt = ω(t)

dω(t)/dt = -3g/2l sin(θ(t)) + 3/ml^2M(t)

Output:

SIR Model

For predicting suspetible, recovered and infected population in a pandemic

Equations

dS(t)/dt = −βS(t)I(t)/N

dI(t)/dt = βS(t)I(t)/N − γI(t)

dR(t)/dt = γI(t),

Output:

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For predicting suspetible, recovered and infected population in a pandemic

Partial Differential Equations

Schrodinger Equation

Equation

i∂ψ(t, x)/∂t =∂^2ψ(t, x)/∂x^2 + V (x)ψ(t, x)

Output:

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Neural ODEs

SIR Model

Solving the SIR model using a Neural Ordinary differential equation to predict infected, susceptible and recoevered population in a sample size of 1000

Equations

dS(t)/dt = −βS(t)I(t)/N

dI(t)/dt = βS(t)I(t)/N − γI(t)

dR(t)/dt = γI(t),

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Neural PDE

1 Dimensional Wave equation

Equations

∂^2u(x, t)/∂t^2 = c^2 ∂^2u(x, t)/∂x^2

u(0, t) = u(1, t) = 0 for all t > 0

(2) u(x, 0) = x(1 − x) for all 0 < x < 1

(3) ∂u(x, 0) ∂t = 0 for all 0 < x < 1

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Universal Differential Equations

Lotka Voltera Predator Prey model

Equations

dx/dt = αx − βxy,
dy/dt = −δy + γxy

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