Nonlinear elliptic system of PDEs

We can also solve nonlinear systems such as the system of nonlinear elliptic PDEs

\[\begin{aligned} \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = uf(\frac{u}{w}) + \frac{u}{w}h(\frac{u}{w}) \\ \frac{\partial^2w}{\partial x^2} + \frac{\partial^2w}{\partial y^2} = wg(\frac{u}{w}) + h(\frac{u}{w}) \\ \end{aligned}\]

where f, g, h are arbitrary functions. With initial and boundary conditions:

\[\begin{aligned} u(0,y) = y + 1 \\ w(1, y) = [cosh(\sqrt[]{f(k)}) + sinh(\sqrt[]{f(k)})]\cdot(y + 1) \\ w(x,0) = cosh(\sqrt[]{f(k)}) + sinh(\sqrt[]{f(k)}) \\ w(0,y) = k(y + 1) \\ u(1, y) = k[cosh(\sqrt[]{f(k)}) + sinh(\sqrt[]{f(k)})]\cdot(y + 1) \\ u(x,0) = k[cosh(\sqrt[]{f(k)}) + sinh(\sqrt[]{f(k)})] \\ \end{aligned}\]

where k is a root of the algebraic (transcendental) equation f(k) = g(k).

This is done using a derivative neural network approximation.

using NeuralPDE, Lux, ModelingToolkit, Optimization, OptimizationOptimJL, Roots
using Plots
import ModelingToolkit: Interval, infimum, supremum

@parameters x, y
Dx = Differential(x)
Dy = Differential(y)
@variables Dxu(..), Dyu(..), Dxw(..), Dyw(..)
@variables u(..), w(..)


# Arbitrary functions
f(x) = sin(x)
g(x) = cos(x)
h(x) = x
root(x) = f(x) - g(x)

# Analytic solution
k = find_zero(root, (0, 1), Bisection())                            # k is a root of the algebraic (transcendental) equation f(x) = g(x)
θ(x, y) = (cosh(sqrt(f(k)) * x) + sinh(sqrt(f(k)) * x)) * (y + 1)   # Analytical solution to Helmholtz equation
w_analytic(x, y) = θ(x, y) - h(k) / f(k)
u_analytic(x, y) = k * w_analytic(x, y)

# Nonlinear Steady-State Systems of Two Reaction-Diffusion Equations with 3 arbitrary function f, g, h
eqs_ = [Dx(Dxu(x, y)) + Dy(Dyu(x, y)) ~ u(x, y) * f(u(x, y) / w(x, y)) + u(x, y) / w(x, y) * h(u(x, y) / w(x, y)),
       Dx(Dxw(x, y)) + Dy(Dyw(x, y)) ~ w(x, y) * g(u(x, y) / w(x, y)) + h(u(x, y) / w(x, y))]

# Boundary conditions
bcs_ = [u(0, y) ~ u_analytic(0, y),
       u(1, y) ~ u_analytic(1, y),
       u(x, 0) ~ u_analytic(x, 0),
       w(0, y) ~ w_analytic(0, y),
       w(1, y) ~ w_analytic(1, y),
       w(x, 0) ~ w_analytic(x, 0)]

der_ = [Dy(u(x, y)) ~ Dyu(x, y),
       Dy(w(x, y)) ~ Dyw(x, y),
       Dx(u(x, y)) ~ Dxu(x, y),
       Dx(w(x, y)) ~ Dxw(x, y)]

bcs__ = [bcs_;der_]

# Space and time domains
domains = [x ∈ Interval(0.0, 1.0),
           y ∈ Interval(0.0, 1.0)]

# Neural network
input_ = length(domains)
n = 15
chain = [Lux.Chain(Dense(input_, n, Lux.σ), Dense(n, n, Lux.σ), Dense(n, 1)) for _ in 1:6] # 1:number of @variables

strategy = QuadratureTraining()
discretization = PhysicsInformedNN(chain, strategy)

vars = [u(x,y),w(x,y),Dxu(x,y),Dyu(x,y),Dxw(x,y),Dyw(x,y)]
@named pdesystem  = PDESystem(eqs_, bcs__, domains, [x,y], vars)
prob = NeuralPDE.discretize(pdesystem , discretization)
sym_prob = NeuralPDE.symbolic_discretize(pdesystem , discretization)

strategy = NeuralPDE.QuadratureTraining()
discretization = PhysicsInformedNN(chain, strategy)
sym_prob = NeuralPDE.symbolic_discretize(pdesystem, discretization)

pde_inner_loss_functions = sym_prob.loss_functions.pde_loss_functions
bcs_inner_loss_functions = sym_prob.loss_functions.bc_loss_functions[1:6]
aprox_derivative_loss_functions = sym_prob.loss_functions.bc_loss_functions[7:end]

callback = function (p, l)
    println("loss: ", l)
    println("pde_losses: ", map(l_ -> l_(p), pde_inner_loss_functions))
    println("bcs_losses: ", map(l_ -> l_(p), bcs_inner_loss_functions))
    println("der_losses: ", map(l_ -> l_(p), aprox_derivative_loss_functions))
    return false
end

res = Optimization.solve(prob, BFGS(); callback = callback, maxiters=5000)

phi = discretization.phi

# Analysis
xs, ys = [infimum(d.domain):0.01:supremum(d.domain) for d in domains]
depvars = [:u,:w]
minimizers_ = [res.u.depvar[depvars[i]] for i in 1:2]

analytic_sol_func(x,y) = [u_analytic(x, y), w_analytic(x, y)]
u_real  = [[analytic_sol_func(x, y)[i] for x in xs for y in ys] for i in 1:2]
u_predict  = [[phi[i]([x,y], minimizers_[i])[1] for x in xs for y in ys] for i in 1:2]
diff_u = [abs.(u_real[i] .- u_predict[i]) for i in 1:2]
for i in 1:2
    p1 = plot(xs, ys, u_real[i], linetype=:contourf, title="u$i, analytic");
    p2 = plot(xs, ys, u_predict[i], linetype=:contourf, title="predict");
    p3 = plot(xs, ys, diff_u[i], linetype=:contourf, title="error");
    plot(p1, p2, p3)
    savefig("non_linear_elliptic_sol_u$i")
end
loss: 184.21893489064263
pde_losses: [0.026450758031687328, 1.481131525234425]
bcs_losses: [2.7626914617818032, 57.39261552109045, 3.7171059615728477, 2.523547528922254, 80.38464468537165, 4.3127655831487415]
der_losses: [2.197228338477056, 24.978105088071626, 0.4582831471567645, 3.9843652917833037]
loss: 145.4145379183263
pde_losses: [1.4527551000787833, 42.5120047535557]
bcs_losses: [1.4535230351306963, 36.41122553312863, 1.958049499782803, 12.737140794556003, 28.455840748758323, 12.07148063691924]
der_losses: [0.5671384118273357, 6.409734215609971, 0.1586278246510645, 1.2270173643277382]
loss: 137.64579163228748
pde_losses: [3.439172478719788, 41.8616355334418]
bcs_losses: [1.8776294118866002, 32.515468433932874, 2.2852216545009116, 11.057360707440381, 30.773109746212313, 10.54659193481632]
der_losses: [0.21610470501431195, 2.4586939540546178, 0.07701320071805606, 0.5377898715494882]
loss: 135.47754810552425
pde_losses: [2.4294265906958286, 40.45504638748009]
bcs_losses: [1.6199009829427553, 34.48794477278722, 2.077579190172519, 10.92087548865285, 30.957013154595654, 10.420843679546223]
der_losses: [0.13898541464981554, 1.5507202664630944, 0.05354316736214114, 0.36566901017605735]
loss: 112.65256881622636
pde_losses: [2.914368036328109, 25.866377182451718]
bcs_losses: [1.2927046421821913, 29.035303954229466, 1.6162577241853242, 7.883919151469181, 30.417362681302922, 7.720614311556389]
der_losses: [0.08393767543721986, 3.9623420055992975, 0.031074893875236962, 1.8283065576092914]
loss: 107.25271184538107
pde_losses: [3.2340193587382857, 23.384025519451345]
bcs_losses: [1.2228742677987605, 28.116033037729302, 1.5254169647422895, 6.9951765631921665, 31.08054880349921, 6.9304004615636225]
der_losses: [0.0894051720467739, 3.4624315704405957, 0.02162172146786394, 1.1907584047108415]
loss: 76.9934040837749
pde_losses: [0.34990440084441155, 8.982250407314275]
bcs_losses: [1.1530458651328994, 30.71139787269078, 0.842010938580868, 3.7452252767097702, 26.15701044212899, 4.032480217487288]
der_losses: [0.049137722458088576, 0.6307060326964573, 0.09607975256696438, 0.2441551551641179]
loss: 62.107593105355
pde_losses: [0.6540290438836015, 5.627920672748485]
bcs_losses: [1.2808275694878377, 25.781669149179898, 0.49756995907571633, 1.9357376528681627, 21.557388187945303, 2.3705297075903253]
der_losses: [0.09847285605701203, 1.6886477940608096, 0.19929870543709427, 0.41550180702076334]
loss: 60.588255545154624
pde_losses: [0.6795200358308091, 5.669222545718874]
bcs_losses: [1.2510369472560543, 25.069120223878926, 0.44997687283159216, 1.9603628324643079, 20.44112532978622, 2.425592116819386]
der_losses: [0.07806193650122656, 1.8563897040567738, 0.15035249689148053, 0.5574945031189724]
loss: 49.091708002840285
pde_losses: [1.0826803286638231, 6.150574768328637]
bcs_losses: [1.2627802770183396, 18.43201563689113, 0.12719145932808232, 2.0589266735578793, 12.685783939179869, 3.0444912601535634]
der_losses: [0.10914628344625006, 2.1540909508780617, 0.8235702166058971, 1.1604562087887529]
loss: 46.48107163129173
pde_losses: [6.447732777587952, 10.364151779264974]
bcs_losses: [0.415679926060219, 9.528981929737913, 0.2264303871399158, 0.24726667046582512, 12.175329285235101, 0.8051887512586178]
der_losses: [0.28254772580798404, 3.279310581856684, 1.0018363895382745, 1.706615427338266]
loss: 44.798170606136836
pde_losses: [9.394584198577117, 8.355546211756804]
bcs_losses: [0.10259218775036595, 7.362421486586367, 0.6393913971219822, 0.3659427264265948, 10.082495912347671, 1.088983075415005]
der_losses: [0.2965413627139235, 3.7007698301368332, 1.213308493649027, 2.1955937236551457]
loss: 39.02796788313464
pde_losses: [10.340990796145318, 5.488727164978546]
bcs_losses: [0.03412812826873523, 6.215165833636941, 1.0624254171331802, 0.1375345853201878, 6.736381523577098, 1.0045857315544526]
der_losses: [0.4634602818905217, 3.944257589341999, 1.4568293256370652, 2.1434815056505974]
loss: 38.9239299127237
pde_losses: [10.673633765279614, 5.193457290938592]
bcs_losses: [0.037588060859493025, 6.183948989388109, 1.0775370114275917, 0.13654801511411113, 6.515151368908171, 1.017754825691886]
der_losses: [0.4244417314689446, 4.019859182233809, 1.4566314174789186, 2.187378253934462]
loss: 37.22199251458557
pde_losses: [8.97866574763996, 5.769099067155842]
bcs_losses: [0.038924135449235224, 5.864297933066895, 1.0672135745538474, 0.19569313581500875, 5.253673131741541, 1.2212860786182491]
der_losses: [0.44199228242524846, 4.418422700395057, 1.518138854702086, 2.454585873022597]
loss: 36.138396876078666
pde_losses: [7.991228667278495, 6.206621822157026]
bcs_losses: [0.045199800219202085, 5.722447506542025, 1.0608352295864951, 0.21805497570460985, 4.804028942911696, 1.3163325295464392]
der_losses: [0.4786470939965828, 4.367210928371277, 1.5476913979981617, 2.380097981766658]
loss: 32.36051274305542
pde_losses: [5.04072366706025, 6.637792004222955]
bcs_losses: [0.06282049549715005, 5.608546368976432, 0.8792633287518747, 0.2066322815640575, 3.937047277414859, 1.3451369671955224]
der_losses: [1.0123052015854532, 3.980477762640286, 1.6913872978210596, 1.9583800903255284]
loss: 30.63717474824997
pde_losses: [4.679257494044196, 6.189351579141561]
bcs_losses: [0.08231440568343973, 5.376566172295877, 0.8648540298201115, 0.1754056174513468, 3.6652564347324144, 1.3204580269435573]
der_losses: [1.1402480516910387, 3.7564283614994824, 1.715657303707366, 1.671377271239574]
loss: 25.719059958245005
pde_losses: [3.195424699375901, 7.657804677676574]
bcs_losses: [0.16135038334988105, 4.1066124932932295, 0.6314400627324938, 0.014915821969216713, 2.452743337405412, 0.9237026067008476]
der_losses: [1.2822513982369288, 2.7132487035447537, 1.7485854318562248, 0.8309803421035431]
loss: 25.305091132579832
pde_losses: [3.089220352675875, 7.482789056606916]
bcs_losses: [0.159860847597183, 4.058627242552206, 0.6142831457970472, 0.012683589158967081, 2.4181874100779877, 0.8862271274381188]
der_losses: [1.3064795233372104, 2.673466168882849, 1.761882423464956, 0.841384244990514]
loss: 19.858452285057826
pde_losses: [0.5293129771093914, 5.820295632378223]
bcs_losses: [0.08120382686547657, 4.077710354992527, 0.41369531905176254, 0.045027720047298596, 1.9726162072909927, 0.5320283301194916]
der_losses: [1.4897780084503465, 2.4242320074850916, 1.7040314111305588, 0.7685204901366666]
loss: 19.789440763412948
pde_losses: [0.7176314906551419, 5.59510177994564]
bcs_losses: [0.0819186856294023, 4.040546240521261, 0.4095184991887973, 0.04626791081248713, 1.9731234769818196, 0.5187225455098952]
der_losses: [1.5073211816621357, 2.4276691352266004, 1.7089550486097305, 0.7626647686700353]
loss: 19.437968768165458
pde_losses: [0.7035755869441692, 5.446162030516845]
bcs_losses: [0.0789865376449236, 3.984111996961838, 0.41000990025155254, 0.048630066910231896, 2.026712095340698, 0.42071416725024624]
der_losses: [1.5247929475956545, 2.373570547762727, 1.6952107761459385, 0.7254921148406326]
loss: 17.787773753025093
pde_losses: [0.6240143841045831, 5.271057723565831]
bcs_losses: [0.135311411613926, 3.2122947113984486, 0.404917554331433, 0.11612678705820081, 1.7605493925298896, 0.1647205707889396]
der_losses: [1.6433482309667453, 1.9268148748111902, 1.5851425208059269, 0.9434755910499786]
loss: 16.923846668081726
pde_losses: [1.4527063333456378, 4.67050606716648]
bcs_losses: [0.2008888947506385, 2.837759249628479, 0.2046631702095295, 0.20618047505758633, 1.5838517882066978, 0.05428809432245977]
der_losses: [1.6819997696297804, 1.6373151058762492, 1.3561469718325188, 1.037540748055667]
loss: 15.811239170551776
pde_losses: [1.0639136492692638, 4.212634850817714]
bcs_losses: [0.27702813722028025, 2.8055563529604925, 0.08220290587294637, 0.1552301863170054, 1.9041342189407309, 0.008130930036499196]
der_losses: [1.6719307017436458, 1.3906986836210535, 1.2534095381388626, 0.9863690156132827]
loss: 14.99562150860077
pde_losses: [0.921417894009466, 3.9371056887591966]
bcs_losses: [0.3288909172154594, 2.451588323741482, 0.09757321549004722, 0.159594666437495, 1.9716878726255294, 0.004513569125229766]
der_losses: [1.6861356121954638, 1.2260606694889804, 1.2230261990339049, 0.9880268804785157]
loss: 13.848995570951843
pde_losses: [1.0584909242845093, 3.5786155783760827]
bcs_losses: [0.32425063657939746, 1.9694108644061419, 0.11921392398806495, 0.15486324050855793, 1.829251826791656, 0.00385501616790109]
der_losses: [1.644795303110908, 1.0546549875305864, 1.1365467140339336, 0.9750465551741035]
loss: 12.115070646308443
pde_losses: [1.0365855146039216, 3.4530065033700823]
bcs_losses: [0.17820026538504144, 1.5611773767642996, 0.16079675634302157, 0.10010709906972999, 1.2108782580331665, 0.0009675320322288005]
der_losses: [1.51020101975591, 0.8905900114557508, 1.0229096209970387, 0.9896506884982521]
loss: 11.170367667677123
pde_losses: [0.8497260371371542, 2.878629403189413]
bcs_losses: [0.1543044550335979, 1.4531373841272865, 0.19760762882106714, 0.05933266779675944, 1.2642590085438024, 0.000838147303008629]
der_losses: [1.4680581807680095, 0.8607745726814129, 1.0392373784006208, 0.9444628038749903]
loss: 10.597129063876702
pde_losses: [0.799904692046703, 2.5458457423851213]
bcs_losses: [0.13963401939021472, 1.3758604937594392, 0.2164217655009833, 0.04681268231571106, 1.224284973527093, 0.00139214274876784]
der_losses: [1.439138019620093, 0.8341314955838062, 1.0333596125622493, 0.9403434244365205]
loss: 9.242329016375614
pde_losses: [0.7737742857526183, 1.7810957657928197]
bcs_losses: [0.32506968476181985, 0.8927721792035348, 0.5407508705753907, 0.028801853299370773, 0.31177893560403536, 0.18881075391851845]
der_losses: [1.2395376560762739, 0.9353346316229194, 1.2974265294007523, 0.9271758703675592]
loss: 9.224342192759563
pde_losses: [0.749984754696115, 1.7873645610768836]
bcs_losses: [0.32436297287605137, 0.8938515095280326, 0.5399129765348994, 0.028746718647734794, 0.31224466228508835, 0.190455982086959]
der_losses: [1.2384945750497698, 0.9343673664536701, 1.3007744535543313, 0.9237816599700268]
loss: 8.619073170174953
pde_losses: [0.4182041077807321, 1.4992391288849822]
bcs_losses: [0.17855806312152672, 1.1705119004661766, 0.4415809188289972, 0.019940194332853836, 0.3821524059323617, 0.2605192173633006]
der_losses: [1.178995349862225, 0.7271273185069327, 1.5128256329201317, 0.8294189321747325]
loss: 6.816664388003794
pde_losses: [0.4652369965570932, 1.181396422324348]
bcs_losses: [0.0844122620942442, 1.3043856644305554, 0.2222563102706794, 0.02845359378183832, 0.35997411552592284, 0.12435909913537224]
der_losses: [0.8299645344847426, 0.47299418733640564, 0.8762256071585035, 0.8670055949040885]
loss: 6.2429057156537695
pde_losses: [0.43173770752293883, 0.9494147373813352]
bcs_losses: [0.07252257913411218, 1.2958900476213975, 0.20109183782343917, 0.04008095043274784, 0.3380926008466482, 0.0901556908130424]
der_losses: [0.7470156001350879, 0.4300017951497805, 0.7452573432211695, 0.9016448255720707]
loss: 5.608656403578268
pde_losses: [0.43022995105595013, 0.8746030831346798]
bcs_losses: [0.056199334434594225, 1.3250790917187052, 0.17255421421984313, 0.04478040599229673, 0.2576449283593392, 0.08401816965908945]
der_losses: [0.6571909755601085, 0.40578539854255224, 0.6339123440177221, 0.6666585068833868]
loss: 5.421955415974468
pde_losses: [0.45258922843594657, 0.8366113796071654]
bcs_losses: [0.0608708705248858, 1.2206588230446929, 0.19499080852085396, 0.03794150488419818, 0.23967763770049735, 0.07907851201382193]
der_losses: [0.642271202218056, 0.391963526950751, 0.6107868305310203, 0.6545150915425777]
loss: 4.658376346786664
pde_losses: [0.6671452358738978, 0.6434564207787586]
bcs_losses: [0.07795430921187067, 1.0840645737314076, 0.11264366381860008, 0.03415068490320878, 0.2396366692113911, 0.019331753662643558]
der_losses: [0.49393835252773477, 0.32389366800526864, 0.4351210607618777, 0.5270399543000039]
loss: 4.3608617967395755
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