Nonlinear hyperbolic system of PDEs

We may also solve hyperbolic systems like the following

\[\begin{aligned} \frac{\partial^2u}{\partial t^2} &= \frac{a}{x^n} \frac{\partial}{\partial x}(x^n \frac{\partial u}{\partial x}) + u f(\frac{u}{w}) \\ \frac{\partial^2w}{\partial t^2} &= \frac{b}{x^n} \frac{\partial}{\partial x}(x^n \frac{\partial u}{\partial x}) + w g(\frac{u}{w}) \\ \end{aligned}\]

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

\[\begin{aligned} u(0,x) &= k * [j_0(ξ(0, x)) + y_0(ξ(0, x))] \\ u(t,0) &= k * [j_0(ξ(t, 0)) + y_0(ξ(t, 0))] \\ u(t,1) &= k * [j_0(ξ(t, 1)) + y_0(ξ(t, 1))] \\ w(0,x) &= j_0(ξ(0, x)) + y_0(ξ(0, x)) \\ w(t,0) &= j_0(ξ(t, 0)) + y_0(ξ(t, 0)) \\ w(t,1) &= j_0(ξ(t, 0)) + y_0(ξ(t, 0)) \\ \end{aligned}\]

where $k$ is a root of the algebraic (transcendental) equation $f(k) = g(k)$, $j_0$ and $y_0$ are the Bessel functions, and $ξ(t, x)$ is:

\[\begin{aligned} \frac{\sqrt[]{f(k)}}{\sqrt[]{\frac{a}{x^n}}}\sqrt[]{\frac{a}{x^n}(t+1)^2 - (x+1)^2} \end{aligned}\]

We solve this with Neural:

using NeuralPDE, Lux, ModelingToolkit, Optimization, OptimizationOptimJL, Roots,
      LineSearches
using SpecialFunctions
using Plots
using DomainSets: Interval
using IntervalSets: leftendpoint, rightendpoint

@parameters t, x
@variables u(..), w(..)
Dx = Differential(x)
Dt = Differential(t)
Dtt = Differential(t)^2

# Constants
a = 16
b = 16
n = 0

# Arbitrary functions
f(x) = x^2
g(x) = 4 * cos(π * x)
root(x) = g(x) - f(x)

# Analytic solution
k = find_zero(root, (0, 1), Roots.Bisection())                # k is a root of the algebraic (transcendental) equation f(x) = g(x)
ξ(t, x) = sqrt(f(k)) / sqrt(a) * sqrt(a * (t + 1)^2 - (x + 1)^2)
θ(t, x) = besselj0(ξ(t, x)) + bessely0(ξ(t, x))                     # Analytical solution to Klein-Gordon equation
w_analytic(t, x) = θ(t, x)
u_analytic(t, x) = k * θ(t, x)

# Nonlinear system of hyperbolic equations
eqs = [Dtt(u(t, x)) ~ a / (x^n) * Dx(x^n * Dx(u(t, x))) + u(t, x) * f(u(t, x) / w(t, x)),
    Dtt(w(t, x)) ~ b / (x^n) * Dx(x^n * Dx(w(t, x))) + w(t, x) * g(u(t, x) / w(t, x))]

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

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

# Neural network
input_ = length(domains)
n = 15
chain = [Chain(Dense(input_, n, σ), Dense(n, n, σ), Dense(n, 1)) for _ in 1:2]

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

@named pdesystem = PDESystem(eqs, bcs, domains, [t, x], [u(t, x), w(t, x)])
prob = discretize(pdesystem, discretization)
sym_prob = 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

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

res = Optimization.solve(prob, BFGS(linesearch = BackTracking()); maxiters = 200, callback)

phi = discretization.phi

# Analysis
ts, xs = [leftendpoint(d.domain):0.01:rightendpoint(d.domain) for d in domains]
depvars = [:u, :w]
minimizers_ = [res.u.depvar[depvars[i]] for i in 1:length(chain)]

analytic_sol_func(t, x) = [u_analytic(t, x), w_analytic(t, x)]
u_real = [[analytic_sol_func(t, x)[i] for t in ts for x in xs] for i in 1:2]
u_predict = [[phi[i]([t, x], minimizers_[i])[1] for t in ts for x in xs] for i in 1:2]
diff_u = [abs.(u_real[i] .- u_predict[i]) for i in 1:2]
ps = []
for i in 1:2
    p1 = plot(ts, xs, u_real[i], linetype = :contourf, title = "u$i, analytic")
    p2 = plot(ts, xs, u_predict[i], linetype = :contourf, title = "predict")
    p3 = plot(ts, xs, diff_u[i], linetype = :contourf, title = "error")
    push!(ps, plot(p1, p2, p3))
end

loss: 44.18193540272127
pde_losses: [39.303473249323986, 0.7345196005703054]
bcs_losses: [0.6082255690867204, 0.35894327772588963, 0.8135484656821607, 0.7178150176553018, 0.8799366219551612, 0.7654736007217434]
loss: 7.573844238452043
pde_losses: [0.02337338713294831, 0.6560297208920118]
bcs_losses: [0.2518349377887783, 1.4915162466463674, 0.39517898386525824, 2.116349031479724, 0.43169769728722107, 2.2078642333597345]
loss: 7.4354118617201514
pde_losses: [0.027864330406965898, 0.19790383797140004]
bcs_losses: [0.2947742326068006, 1.5401151031009628, 0.4476271850453586, 2.1728127315557484, 0.48797305468161234, 2.266341386351304]
loss: 4.933639641434781
pde_losses: [0.015188002803257014, 0.25970907025464407]
bcs_losses: [0.1191819334034509, 0.9265236119081992, 0.26292610308881376, 1.5214596689510378, 0.2510068475667955, 1.577644403458582]
loss: 2.787287087037753
pde_losses: [0.15312839609738785, 2.0833424380689856]
bcs_losses: [0.02054167570434923, 0.00677996593804302, 0.1243426937866623, 0.16681646980158893, 0.09187115964600627, 0.14046428799472996]
loss: 1.792695476789706
pde_losses: [0.14985778197616537, 1.0484519716405558]
bcs_losses: [0.02867893822775809, 0.00800083962710128, 0.1430593011686944, 0.17033981824553096, 0.10824751313310321, 0.13605931277079708]
loss: 1.120697507443622
pde_losses: [0.14566607121841527, 0.2802092610688613]
bcs_losses: [0.04508529946546634, 0.01125277057855251, 0.17642799146438948, 0.18270211179192664, 0.13805101984214047, 0.14130298201387004]
loss: 0.7839284813217092
pde_losses: [0.049797166936214346, 0.052702291577365015]
bcs_losses: [0.07315824019808212, 0.0028853408241223647, 0.20442208445557283, 0.1409106278392264, 0.1666195396309177, 0.09343318986020849]
loss: 0.6335754439830951
pde_losses: [0.022446334739708678, 0.056438140309193016]
bcs_losses: [0.06705476625569534, 0.003596437321553687, 0.18292147559736593, 0.09796612475831888, 0.1507561622202182, 0.052396002781041284]
loss: 0.5931275975416709
pde_losses: [0.01858030435541553, 0.04957539702142281]
bcs_losses: [0.06369225530191676, 0.005784626268534087, 0.175086486408892, 0.09091583357871934, 0.14515348162495226, 0.04433921298181808]
loss: 0.5349985160418816
pde_losses: [0.012497787157720437, 0.053976002052684796]
bcs_losses: [0.05395038735442378, 0.009065404350636456, 0.15628624377684264, 0.08374330700083764, 0.1293602067273668, 0.03611917762136905]
loss: 0.4933728797093528
pde_losses: [0.00751317884103414, 0.06725149679453664]
bcs_losses: [0.045121475757061956, 0.011152608166245102, 0.13867392472304105, 0.07825909508565597, 0.11377900606994855, 0.03162209427182937]
loss: 0.4542813011946394
pde_losses: [0.0034419743997987184, 0.08156743985249489]
bcs_losses: [0.0363343036094757, 0.012541214940378517, 0.11997252575627855, 0.07438747754887337, 0.09757876272462562, 0.02845760236271405]
loss: 0.4296351583228248
pde_losses: [0.0018552167460915347, 0.08956002070325758]
bcs_losses: [0.031598106641306446, 0.014428840570787348, 0.10964140340079806, 0.06819089243978663, 0.08880936750320617, 0.025551310317591006]
loss: 0.3877426316512764
pde_losses: [0.0005922267909847108, 0.0906047777335937]
bcs_losses: [0.025594276965143763, 0.01869151274646866, 0.0963731227975984, 0.05714801889567366, 0.07778205262875769, 0.020956643093055817]
loss: 0.3078661970118107
pde_losses: [0.0005883391056022985, 0.0798061443772949]
bcs_losses: [0.015595721997843075, 0.032709569761242716, 0.0734652509241587, 0.03248886740873364, 0.05862075809911094, 0.014591545337824481]
loss: 0.225186985785587
pde_losses: [0.003565306164201245, 0.049028900873011226]
bcs_losses: [0.00344797942280638, 0.07074984800276535, 0.039914149624693335, 0.009458626813274917, 0.029711389548599892, 0.01931078533623461]
loss: 0.1833441400063946
pde_losses: [0.004614497253082752, 0.03453502740502643]
bcs_losses: [0.0007607278348143251, 0.07100286904775133, 0.026619942655270165, 0.008369840479462473, 0.018708183219423524, 0.018733052111563616]
loss: 0.14908030098298747
pde_losses: [0.004131985835321704, 0.03229583487039831]
bcs_losses: [0.00023025736098614458, 0.060402406756826045, 0.01803620983586786, 0.007486904321254579, 0.012291498449187525, 0.014205203553145298]
loss: 0.0663130594956842
pde_losses: [0.001039421783978474, 0.009424139632897658]
bcs_losses: [0.00025947454045746393, 0.01581054542151811, 0.01332256224912393, 0.006598364451218887, 0.009957835068831085, 0.009900716347658585]
loss: 0.05260806284859603
pde_losses: [0.0005399412662374499, 0.004532894553108336]
bcs_losses: [0.0004480910428691925, 0.012419367473515344, 0.010627403135943842, 0.004796100908816026, 0.008230493079252641, 0.0110137713888532]
loss: 0.04408842857959918
pde_losses: [0.0004731057543620435, 0.00232998112750284]
bcs_losses: [0.000997701072969565, 0.010947821440094567, 0.007853077065476681, 0.003623432385035127, 0.006286116127538545, 0.01157719360661981]
loss: 0.0373020184717082
pde_losses: [0.002628764874850996, 0.0033160309797347744]
bcs_losses: [0.003100314220875649, 0.00690701658801827, 0.003404449551563178, 0.00210637269443821, 0.0034386451871590037, 0.012400424375068112]
loss: 0.03272706186274966
pde_losses: [0.0015623006407507257, 0.0010404909373581]
bcs_losses: [0.0029979412287133218, 0.00891447867362405, 0.0037871849144191947, 0.002199537026189285, 0.003585572812266973, 0.008639555629428012]
loss: 0.030478596945482453
pde_losses: [0.0012632321024056003, 0.0017656865527727868]
bcs_losses: [0.0024234738071643577, 0.0070876866042117155, 0.004156203438984433, 0.001674283390360964, 0.003809404527186463, 0.008298626522396133]
loss: 0.024463353047207634
pde_losses: [0.0011304113373154872, 0.0033661661875021695]
bcs_losses: [0.0021452126742379386, 0.0035758915414286743, 0.003553798104697041, 0.0008893490420120671, 0.003104463857202128, 0.006698060302812128]
loss: 0.018310431326517416
pde_losses: [0.003481466000977417, 0.003531508079543008]
bcs_losses: [0.0013293365803360123, 0.0008656887947270334, 0.0024893235368493583, 0.001250597064985795, 0.0017270959337714546, 0.003635415335327338]
loss: 0.016097671184535596
pde_losses: [0.0030427646003575023, 0.002632913821028831]
bcs_losses: [0.001004692291675647, 0.0010778766252839153, 0.0023379447996142704, 0.0014518318605230666, 0.0016587560600960153, 0.00289089112595635]
loss: 0.014138244319796942
pde_losses: [0.002358635174993572, 0.002475110412299104]
bcs_losses: [0.0008853145396850332, 0.0011964522525974523, 0.0019509824531387862, 0.0014175733438128125, 0.0015285610164212289, 0.0023256151268489533]
loss: 0.010387368721769527
pde_losses: [0.0016526428404804578, 0.001944277931158913]
bcs_losses: [0.0006040033270363759, 0.0010285182871186327, 0.0012196252015920098, 0.0018494429227673271, 0.0011684793655426795, 0.0009203788460731313]
loss: 0.006863698078731401
pde_losses: [0.001502845584039318, 0.001190185599790228]
bcs_losses: [0.00011240378326142973, 0.00012574381812091058, 0.000855508921041023, 0.0013969113305035928, 0.0009672441914474531, 0.0007128548505274451]
loss: 0.005965667161005871
pde_losses: [0.0014302726096179586, 0.0010540245623794572]
bcs_losses: [0.00015534850300250418, 0.00010421773269787248, 0.0006644772250376368, 0.0011401031188877136, 0.0007267174615287447, 0.0006905059478539836]
loss: 0.0055181758543149835
pde_losses: [0.0010862382523290093, 0.0011495644697433777]
bcs_losses: [8.947333888794352e-5, 0.00021021189156909397, 0.0008871654474978754, 0.0006758320857671774, 0.0008706248995963174, 0.0005490654689241886]
loss: 0.004872670237564871
pde_losses: [0.0009616691130007001, 0.0006329456722576398]
bcs_losses: [7.545299137609365e-5, 0.00027051202282708055, 0.000848800362082993, 0.0006537252021350026, 0.0008860739349664637, 0.0005434909389188983]
loss: 0.00448968008720989
pde_losses: [0.0008365321962080613, 0.0005621951395802474]
bcs_losses: [0.00010468239363950203, 0.0003255023272150419, 0.0006426322621259604, 0.0007037543350427296, 0.0007179035121274083, 0.0005964779212709393]
loss: 0.004002908740312231
pde_losses: [0.0005641864729636481, 0.0006479176144475188]
bcs_losses: [0.00011601817567336725, 0.0004020419623458435, 0.0004981537394332554, 0.0005946321731551217, 0.0005839485356599692, 0.0005960100666335071]
loss: 0.003455208485770081
pde_losses: [0.00027134181175388577, 0.0007976088683219979]
bcs_losses: [0.00017543460924671414, 0.0004335999151720785, 0.00035900903748731493, 0.000505269737666087, 0.0004473772895907777, 0.0004655672165312248]
loss: 0.0032863303044192264
pde_losses: [0.00021492769289958006, 0.0008798257035252411]
bcs_losses: [0.00021283436244796093, 0.00038126790663480485, 0.0002933620309007318, 0.00048735911379212284, 0.00038819264967655933, 0.00042856084454222557]
loss: 0.003155323626271824
pde_losses: [0.00017025929458325075, 0.0009729288292691403]
bcs_losses: [0.00026012190159320714, 0.00031851594884889996, 0.0002348646887565939, 0.00045390714873023644, 0.000342319602606372, 0.00040240621188412386]
loss: 0.002991123129551109
pde_losses: [0.00012791638337046358, 0.0009780710888050313]
bcs_losses: [0.00028250427426112937, 0.00024157918062866511, 0.00019850075431096054, 0.0004245318800595347, 0.0003250443902276749, 0.00041297517788764943]
loss: 0.002779139683506391
pde_losses: [9.427148213590088e-5, 0.0008614430123242376]
bcs_losses: [0.00028994201610431624, 0.00018219393101637677, 0.00016877544703877568, 0.0004020213986259905, 0.0003217608084633698, 0.00045873158779742356]
loss: 0.002523983849824823
pde_losses: [0.00010201674345655434, 0.0006040945119108497]
bcs_losses: [0.00023296970078255929, 0.000145847099657471, 0.0001648714860165639, 0.00041360566218472204, 0.00035375683643246506, 0.0005068218093836377]
loss: 0.002324859323539378
pde_losses: [0.0001763422136628676, 0.0003722501956632099]
bcs_losses: [0.00018416203893627725, 0.0001646441241478385, 0.00015966619752987493, 0.00040837642650556275, 0.0003741789306760936, 0.00048523919641765357]
loss: 0.002255859944204344
pde_losses: [0.00023012211254941933, 0.00024644493905990374]
bcs_losses: [0.00011208493884416356, 0.00019728573155867215, 0.00019842846598710682, 0.00039607972849137093, 0.0004328741858156444, 0.0004425398418980631]
loss: 0.0022085924705952343
pde_losses: [0.0002553315714127453, 0.00024111118680182]
bcs_losses: [0.0001142856220752123, 0.00022598073440835865, 0.00018616793085988748, 0.0003696451913570773, 0.0003990200333212716, 0.00041705020035886196]
loss: 0.002189818442209597
pde_losses: [0.00023159493981926604, 0.0002495484332550293]
bcs_losses: [0.00011730741534759115, 0.00022965608133779445, 0.00018869002911420062, 0.00036425654537989427, 0.00039494811198263166, 0.0004138168859731893]
loss: 0.002139363594936813
pde_losses: [0.0001903549261197845, 0.00029184251200113144]
bcs_losses: [0.00011853990725138754, 0.00022774684600364572, 0.00018179574901289088, 0.0003567626767349936, 0.0003672781654275229, 0.0004050428123854564]
loss: 0.0020708804317625114
pde_losses: [0.0001522799552997647, 0.0003214770126417391]
bcs_losses: [0.000123991498410531, 0.0002260679709097096, 0.00016965462677042087, 0.0003394212574786244, 0.000343806150127018, 0.00039418196012470377]
loss: 0.0019213400768094244
pde_losses: [0.00011464077076645511, 0.0003787974583960625]
bcs_losses: [0.0001260678918966256, 0.0002192103783399802, 0.00012910634661655984, 0.0003057974790751095, 0.0002800830402626799, 0.00036763671145595173]
loss: 0.001753008491771853
pde_losses: [8.034867943595358e-5, 0.00038196932231873704]
bcs_losses: [0.00013839051217987513, 0.0002079820360826389, 9.965586174494856e-5, 0.0002517067543898635, 0.00023863340223853705, 0.00035432192338129937]
loss: 0.0016303268514898223
pde_losses: [7.368093289819489e-5, 0.0003557358770590684]
bcs_losses: [0.00013046729737729198, 0.00019130286862725424, 9.344788984694193e-5, 0.00021167096889683, 0.00021623102727473718, 0.00035778998950950375]
loss: 0.0015690535499771805
pde_losses: [7.231585969266675e-5, 0.00028424533207900636]
bcs_losses: [0.00010398676193690688, 0.00016769740527138982, 9.318260728449171e-5, 0.00019993742794260414, 0.00025341254530003853, 0.0003942756104700762]
loss: 0.0015458486671927761
pde_losses: [8.017143406352623e-5, 0.0002668210436749956]
bcs_losses: [9.423423216173991e-5, 0.000154087668803643, 9.793014142635176e-5, 0.0001980813334533392, 0.00025073795697117197, 0.0004037848566380085]
loss: 0.001525314648494433
pde_losses: [8.292590789233563e-5, 0.00024378510722787317]
bcs_losses: [8.394611474789761e-5, 0.00015910700674505362, 9.948148929359765e-5, 0.00020032699615970748, 0.0002612837831431607, 0.000394458243284807]
loss: 0.0015016690890782753
pde_losses: [7.503386550391388e-5, 0.0002037389084462372]
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ps[1]

ps[2]