Ref Bari
Advisor: Prof. Brendan Keith
Elliptical Orbits
Learning Rate
Circular Orbits
Elliptical Orbits
Learning Rate
Circular Orbits
Elliptical Orbits
Learning Rate
Circular Orbits
Circular Orbits
Epochs
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Exact Schwarzschild
Hamiltonian
Geodesic Equations of Motion
Integrate Equations to get Orbit
Convert Orbit to Waveform
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Exact Schwarzschild Hamiltonian
Geodesic Equations of Motion
Integrate Equations to get Orbit
Convert Orbit to Waveform
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Define Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
end
Define Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
end
Define Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
end
Define Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
end
Define Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
end
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Define Callback Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
losses = []
function callback(pn, loss; dotrain = true)
if dotrain
push!(losses, loss);
@printf("Epoch: %d, Loss: %15.12f \n",length(losses),loss);
p = plot(losses, label = "Loss")
display(p)
else
prinln(l)
end
return false
end
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Predicted Hamiltonian
Equations of Motion
Predicted Waveform
Compare to True Waveform
Adjust Hamiltonian
Keep Training Network
Predicted Hamiltonian
Equations of Motion
Predicted Waveform
Compare to True Waveform
Adjust Hamiltonian
Keep Training Network
Learning orbital dynamics of binary black hole systems from gravitational wave measurements (B. Keith et. al., 2021)
Anatomy of a Black Hole
Anatomy of a Black Hole
Schwarzschild Radius
Photon Ring
Innermost Stable Circular Orbit
Schwarzschild Radius
Photon Ring
Innermost Stable Circular Orbit
✅
✅
But we want this in terms of p and e ...
For a circular orbit ...
For a circular orbit:
For a circular orbit:
Kinetic Energy
Fixed Constant Energy of Particle
Gravitational Potential created by Black Hole
radius is constant for a circular orbit!
For a circular orbit:
Why does this term keep showing up?
Ah ... it's a sign of the event horizon of the black hole!
The metric becomes singular at the event horizon of the black hole!
The metric becomes singular at the event horizon of the black hole!
The metric becomes singular at the event horizon of the black hole!
The metric becomes singular at the event horizon of the black hole!
The metric becomes singular at the event horizon of the black hole!
The metric becomes singular at the event horizon of the black hole!
The metric becomes singular at the event horizon of the black hole!
What if we had a circular orbit?
What if we had a circular orbit?
Time slows down dramatically near p=6! Why?
Time slows down dramatically near p=6! Why?
Time stops at the event horizon!
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Stable Circular Orbit
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Elliptical Orbit
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Hyperbolic Orbit
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Unstable Circular Orbit
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Plunge Orbit
Circular
Elliptical
Hyperbolic
Plunge
Elliptical
Hyperbolic
Plunge
Circle
Elliptical
Hyperbolic
Plunge
Circle
Elliptical
Hyperbolic
Plunge
Circle
Hyperbolic
Plunge
Circle
Ellipse
Circle
Ellipse
Ellipse
Unbound
Circle
Ellipse
Ellipse
Constraints:
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Mission Statement
Introduction to GENERIC
G
E
N
E
R
I
C
Introduction to GENERIC
G
E
N
E
R
I
C
eneral
quation for
on
quilibrium
eversible
rreversible
oupling
Introduction to GENERIC
Introduction to GENERIC
"reversible"
"irreversible"
Introduction to GENERIC
"reversible"
"irreversible"
Introduction to GENERIC
"reversible"
"irreversible"
Introduction to GENERIC
Equilibrium
Inequilibrium
Introduction to GENERIC
Equilibrium
Inequilibrium
Introduction to GENERIC
Equilibrium
Inequilibrium
Conservative Dynamics
Introduction to GENERIC
Equilibrium
Inequilibrium
Dissipative Dynamics
Introduction to GENERIC
Equilibrium
Inequilibrium
Introduction to GENERIC
Equilibrium
Inequilibrium
Introduction to GENERIC
Equilibrium
Inequilibrium
Examples of GENERIC
Examples of GENERIC
State Variables
State Variables
Examples of GENERIC
State Variables
Energy
Energy
Examples of GENERIC
State Variables
Energy
Energy
Entropy
Energy
Energy
Entropy
Examples of GENERIC
State Variables
Energy
Energy
Entropy
Entropy
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC
Examples of GENERIC