(9)) has the form 8 > > > > > < > > > > > : t 2 + H (x, Dx (x, t), ⇢(x, t), t) = 0 ⇢t 2 ⇢ divx (⇢DpH (x, Dx (x, t), ⇢(x, t), t)) = 0 ⇢ 0, R Td ⇢(x, t)dx = 1, t 2 [0, T] ⇢(x, T) = ⇢T (x), (x, 0) = 0(x, ⇢(x, 0)), x 2 Td . (10) Given: H : Td ⇥ Rd ⇥ X ⇥ R ! R is a periodic in space and convex in momentum Hamiltonian, where X = R+ or X = L 1(Td ; R+) or X = R+ ⇥ L 1(Td ; R+). H = L ⇤ is a di↵usion parameter 0 : Td ⇥ X ! R, ⇢T : Td ! R given initial-terminal conditions Unknowns: , ⇢ : Td ⇥ [0, T] ! R has the form 5
satisﬁes the following equations and constraints. @t⇢S + r · mS = ⇢S K ⇤ ⇢I + ⌘2 S 2 ⇢S ✓1⇢V ⇢S @t⇢I + r · mI = ⇢S K ⇤ ⇢I ⇢I + ⌘2 I 2 ⇢I @t⇢R + r · mR = ⇢I + ⌘2 R 2 ⇢S + ✓1⇢V ⇢S PDE of the vaccine distribution @t⇢V = f(t, x) ✓1⇢V ⇢S , 0 t < T 0 @t⇢V + r · mV = ✓2⇢V ⇢S , T 0 t T and constraints on the vaccine density function ⇢V and the factory function f(t, x). 0 f(t, x) fmax (t, x) 2 [0, T 0] ⇥ ⌦factory f(t, x) = 0 (t, x) 2 [0, T 0] ⇥ ⌦\⌦factory ⇢V (t, x) Cfactory (t, x) 2 [0, T 0] ⇥ ⌦factory 4 Vaccine model 9
Z ⌦ X i=S,I,R Fi(⇢i, mi)dx dt + Z T T 0 Z ⌦ FV (⇢V , mV )dx dt + Z T 0 GP ((⇢S + ⇢I + ⇢R)(t, ·)) + GV (⇢V (t, ·))dt + Z T 0 0 G0(f(t, ·))dt Here, the cost functional satisﬁes the following conditions. I minimize the transportation cost for moving each population; I minimize the total number of infected people and susceptible people by maximizing the usage of the vaccines at time T ; I maximize the total number of recovered people at time T ; I avoid high concentration of population and vaccines at each time t 2 (0, T ); I minimize the amount of vaccines produced during t 2 (0, T 0); I minimize the transportation cost for delivering vaccines during t 2 (T 0, T ). 5 10
adjusted to model other natural disasters. I Rescue/Control problems in California wildﬁres; I Evaluation plans for Florida hurricanes, e.t.c. References: I Lee, W., Liu, S., Li, W., Osher, S. (2021). Mean ﬁeld control problems for vaccine distribution. arXiv:2104.11887. 9 Evacuation 14
Mean Field Planning Jiajia Yu1 Rongjie Lai1 Wuchen Li2 Stanley Osher3 June 7, 2021 1Department of Mathematics, Rensselaer Polytechnic Institute, United States. 2Department of Mathematics, University of South Carolina, United States. 3Department of Mathematics, University of California, Los Angeles, United States. Funding: J. Yu and R. Lai’s work are supported in part by an NSF career award DMS–1752934. W. Li and S. Osher’s work are supported in part by AFOSR MURI FP 9550-18-1-502. 15
rx · m = 0, ⇢(0, ·) = ⇢0, ⇢(1, ·) = ⇢1} . I Optimal Transport min ⇢,m2C(⇢0,⇢1) Z 1 0 Z ⌦ km(t, x)k2 2 2⇢(t, x) dxdt. I Mean Field Planning min ⇢,m2C(⇢0,⇢1) Z 1 0 Z ⌦ km(t, x)k2 2 2⇢(t, x) + F(x, ⇢(t, x))dxdt. Let C(⇢0) := {(⇢, m) : @t⇢ + rx · m = 0, ⇢(0, ·) = ⇢0}. I Mean Field Game min ⇢,m2C(⇢0) Z 1 0 Z ⌦ km(t, x)k2 2 2⇢(t, x) + F(x, ⇢(t, x))dxdt + Z ⌦ G(x, ⇢(1, x))dx. 3 17
for MFP: I The gradient descent step is easy to compute as long as regularizer F is di↵erentiable. I The proximal descent step is independent to the form of F. PGD is also e cient: I The gradient descent step can be implemented elementwise. I The proximal descent step is e cient to compute with fast cosine transform. I If ⌦ = [0, 1]d 1 and the domain [0, 1] ⇥ ⌦ is uniformly discretized to nd grids, then the computation cost for each step is O(dnd log n). I By applying accelerated PGD and multigrid strategies, we can further reduce the number of iterations and total time for convergence. 5 19
Mk) as an approximation of (⇢k, mk) on grid points. With mild assumptions, We, for the ﬁrst time, establish the following convergence based on an optimization algorithm: Theorem (Yu-Lai-Li-Osher’21) I (Pk, Mk) ! (⇢k, mk) on grid points for any ﬁxed k as mesh size h ! 0. I (Pk, Mk) ! (⇢⇤, m⇤) on grid points as h ! 0, k ! 1. I (P⇤, M⇤) ! (⇢⇤, m⇤) on grid points as h ! 0 6 20
0 Z ⌦ km(t, x)k2 2 2⇢(t, x) dxdt Table: E ciency and accuracy comparison of OT with best performance highlighted in red. (ALG=Augmented Lagrangian, G-prox=G-prox PDHG. Left: 1D. Right: 2D. nt =64, nx = ny =256.) 7 21
Z ⌦ km(t, x)k2 2 2⇢(t, x) + Q⇢(t, x)Q(x)dxdt where Q(x) = ( 1, x 2 white region, 0, otherwise , . Figure: Evolution of densities for various ⇢0, ⇢1, Q(x). 8 22
equilibrium solution can be found by minimizing an overall “energy” (e.g. multiply the value function for a single agent by ⇢): A(⇢, v) = min ⇢,v Z T 0 Z ⌦ ⇢(x, t)L(v) + F(x, ⇢) dx dt + Z ⌦ g(x)⇢(x, T) dx where x = x(t), v = v(t) above, and where the optimization has the constraint, @t⇢ ⌫ ⇢ + div(⇢v) = 0. 2 26
constraints into the objective, integrating by parts, and calculating the Legendre transform, we get = max ' min ⇢ Z T 0 Z ⌦ F(x, ⇢(x, t)) dx dt + Z ⌦ ⇣ g(x) '(x, T) ⌘ ⇢(x, T) dx + Z ⌦ '(x, 0)⇢(x, 0) dx + Z T 0 Z ⌦ ✓ @t'(x, t) + '(x, t) L⇤( r'(x, t)) ◆ ⇢(x, t) dx dt which means we end up with a sampling problem – this is a preview of APAC-Net. This is in the spirit of Feynman-Kac. Then the idea is to turn ⇢ and ' into neural networks and train as in GANs (Generative Adversarial Networks). 3 27
I A 20 dimensional obstacle problem where we have an obstacle in (x1, x2) and in (x3, x4) and in (x5, x6), etc. Congestion penalty is active in the bottlenecks. Figure: A screencapture of a video that will be played 6 30
a quadcopter are: 8 > > > > > > < > > > > > > : ¨ x = u m (sin( ) sin( ) + cos( ) cos( ) sin(✓)) ¨ y = u m ( cos( ) sin( ) + cos( ) sin(✓) sin( )) ¨ z = u m cos(✓) cos( ) g ¨ = ˜ ⌧ ¨ ✓ = ˜ ⌧✓ ¨ = ˜ ⌧ where u is the thrust, g is the gravitational acceleration (9.81m/s2), and x, y, z are the spatial coordinates, , ✓, are the angular coordinates, and ˜ ⌧ , ˜ ⌧✓ , ˜ ⌧ . Turns 12-dimensional when you transfer to ﬁrst-order system. 7 31