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 ModelingToolkit, NeuralPDE, SciMLBase, Lux, Optimization, OptimizationOptimJL, NonlinearSolve,
      LineSearches
using Optim: BFGS
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, p) = g(x) - f(x)

# Analytic solution
k = solve(IntervalNonlinearProblem(root, (0.0, 1.0)), ITP()).u  # 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 = LineSearches.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: 4.7215889016917725
pde_losses: [0.4067883986046623, 3.9063816490939747]
bcs_losses: [0.13750448495741519, 0.11019781513743625, 0.059931015506652585, 0.022543516045638798, 0.05725693636531878, 0.020985085980673193]
loss: 1.697917249868058
pde_losses: [0.15500368908574694, 1.2387331516724516]
bcs_losses: [0.024141475110986917, 0.17924184580136565, 0.004846545028145847, 0.049289667836190425, 0.005072557330131199, 0.041588318003039366]
loss: 1.0589983268240077
pde_losses: [0.08994692103165666, 0.7817598572403426]
bcs_losses: [0.0008525646798425979, 0.0005920191957144533, 0.0307527550132442, 0.054584832993073706, 0.032238792923443144, 0.0682705837466904]
loss: 1.0061594504132425
pde_losses: [0.3075385724148408, 0.3089972313455718]
bcs_losses: [0.04077738632339273, 0.01653847011073735, 0.009599783724108853, 0.14088017327806246, 0.021676466076600863, 0.16015136713992784]
loss: 0.8981936083244524
pde_losses: [0.19313005252560342, 0.345124720719843]
bcs_losses: [0.0409877308659299, 0.012166275406763622, 0.009855989478015104, 0.129163590063463, 0.022159206891963955, 0.14560604237287036]
loss: 0.5819666190940604
pde_losses: [0.10089909925714034, 0.10600868194396185]
bcs_losses: [0.029010115199517406, 0.018444619475441907, 0.005381310146111519, 0.1630320410252563, 0.014203172500767938, 0.1449875795458632]
loss: 0.511490007678982
pde_losses: [0.057577254591811525, 0.054826115004164414]
bcs_losses: [0.02295801115530236, 0.02524837611878917, 0.0027097392448531096, 0.17987809242321381, 0.010450481913458378, 0.15784193722738926]
loss: 0.45190828641482433
pde_losses: [0.018190565792032622, 0.04087096333772982]
bcs_losses: [0.02603842997080691, 0.024777209690198546, 0.002973234478515691, 0.17745117428911467, 0.012791037861538594, 0.14881567099488752]
loss: 0.4116692697806785
pde_losses: [0.013407864332987966, 0.003213657296522187]
bcs_losses: [0.03634927670396185, 0.023136068164623033, 0.005646505068996314, 0.17399142973366682, 0.020327928193927026, 0.1355965402859933]
loss: 0.39839337968033967
pde_losses: [0.005256754659458795, 0.004701870445223925]
bcs_losses: [0.039839904252329215, 0.0207286164145213, 0.006742199173679653, 0.16764939567477088, 0.02392118735483433, 0.12955345170552157]
loss: 0.3893990617518537
pde_losses: [0.004714197925260462, 0.005814173232321669]
bcs_losses: [0.04793214810670777, 0.01728640824105764, 0.009689028221217104, 0.15536162007380677, 0.030083423627752567, 0.11851806232372968]
loss: 0.3809400191228773
pde_losses: [0.004819380807838239, 0.004757455972444836]
bcs_losses: [0.05822912098014433, 0.01313122062752457, 0.015744137047105096, 0.13912110792236126, 0.038593430222515705, 0.10654416554294323]
loss: 0.3670898697287058
pde_losses: [0.0049110849811263915, 0.015090717099202195]
bcs_losses: [0.06470909767388981, 0.008061431884604906, 0.026154266398786454, 0.11355798588237141, 0.04484815937236293, 0.08975712643636166]
loss: 0.359215810777263
pde_losses: [0.005029758725039993, 0.01380599320469135]
bcs_losses: [0.06594729052630624, 0.006995821983931845, 0.03070691717037081, 0.10522142054768792, 0.04625883677328567, 0.08524977184594919]
loss: 0.3519690753171955
pde_losses: [0.005701350445676413, 0.015789524168420197]
bcs_losses: [0.06443074431999216, 0.006459210516447323, 0.03327884085660195, 0.09859562536667779, 0.045264059502747346, 0.08244972014063236]
loss: 0.3416311899162344
pde_losses: [0.006884418716549875, 0.019325150084129505]
bcs_losses: [0.0601598205360398, 0.006145851351847394, 0.035074925430195907, 0.0921252705620351, 0.04201093206032418, 0.07990482117511269]
loss: 0.31929907318173134
pde_losses: [0.01499325074187664, 0.03263359481607408]
bcs_losses: [0.04664914911490125, 0.00558364370220636, 0.036694091250128194, 0.07662971306000661, 0.03170857821103832, 0.07440705228549989]
loss: 0.3113649501031357
pde_losses: [0.016940863005761385, 0.037308965979021905]
bcs_losses: [0.04081109858143862, 0.005815275562142066, 0.035726169412602884, 0.07406397251397845, 0.027238489392751794, 0.0734601156554386]
loss: 0.3029271226515058
pde_losses: [0.019163950390733507, 0.040049451680079555]
bcs_losses: [0.03563788620146793, 0.005996791516874728, 0.03433115113807967, 0.07166560499386121, 0.023338879656073258, 0.07274340707433591]
loss: 0.2888801241491686
pde_losses: [0.01859792162140066, 0.04202568992215557]
bcs_losses: [0.02927666665464565, 0.0062894634152138055, 0.032023681190947915, 0.07016668251942867, 0.018817260387189404, 0.07168275843818692]
loss: 0.27121562396410337
pde_losses: [0.01787422648615628, 0.045379277968298236]
bcs_losses: [0.022773213445653383, 0.005998929939258252, 0.02980225399762252, 0.06669300595903999, 0.014595635295277106, 0.06809908087279755]
loss: 0.2478227191358567
pde_losses: [0.012069179216207877, 0.04953177383679999]
bcs_losses: [0.016374377575964036, 0.005397970588441152, 0.026573416888212912, 0.06432403832232349, 0.010710965851018545, 0.06284099685688872]
loss: 0.2184588276829028
pde_losses: [0.004067261520927185, 0.06006242737050211]
bcs_losses: [0.00946045984401319, 0.0030371658660219283, 0.02028264409300919, 0.06449100438424589, 0.00749617341968065, 0.04956169118450269]
loss: 0.20232792392911597
pde_losses: [0.004207302078700552, 0.040905924210600166]
bcs_losses: [0.010906132702601592, 0.0023464700034256246, 0.017865970678623232, 0.06744336513750111, 0.00801269409201761, 0.05064006502564611]
loss: 0.1749767838043249
pde_losses: [0.00880983833184923, 0.05268370512480178]
bcs_losses: [0.01791717213835948, 0.008787248340121546, 0.012191963453832434, 0.038641276151323366, 0.010670638432657243, 0.025274941831379846]
loss: 0.1632939767858643
pde_losses: [0.015441095202641981, 0.01602095396727532]
bcs_losses: [0.02854197017417893, 0.02435668773324529, 0.009686285684602424, 0.032251640474410884, 0.014299440643822442, 0.022695902905687054]
loss: 0.1564777202718346
pde_losses: [0.011052143479506045, 0.039845886410934055]
bcs_losses: [0.013506905284470792, 0.017250264955043573, 0.001693810917626897, 0.039601175370315815, 0.004030357485976464, 0.029497176367960957]
loss: 0.14165794871798237
pde_losses: [0.0038974132922487094, 0.012877929083296939]
bcs_losses: [0.006125751671342444, 0.001105727460965696, 0.0006464484098150639, 0.06286288848027774, 0.0012163155759766825, 0.05292547474405909]
loss: 0.1125219462773584
pde_losses: [0.00300920173348097, 0.00922022867065794]
bcs_losses: [0.015357178563373982, 0.011735070550064003, 0.004287032869927992, 0.035321401181898554, 0.005436937497692528, 0.02815489521026244]
loss: 0.10231373464858794
pde_losses: [0.002225392173783172, 0.0061535569661728285]
bcs_losses: [0.014876533455360115, 0.012515358855740644, 0.003846432693181872, 0.03200254108513525, 0.0050932411609188955, 0.025600678258295158]
loss: 0.07183881745143991
pde_losses: [0.0015368600706842702, 0.02299572931925268]
bcs_losses: [0.0087055787276527, 0.013959917566984361, 0.0017901418013609432, 0.009683701847062815, 0.0019167990175308167, 0.011250089100911326]
loss: 0.059021719771843964
pde_losses: [0.003863033765319524, 0.018231006765031626]
bcs_losses: [0.004943858687883985, 0.0055081391824213875, 0.0012937145676545658, 0.009780846086773789, 0.0007344123344509438, 0.014666708382308139]
loss: 0.04144001276649116
pde_losses: [0.0038349008742654665, 0.0037724771838366076]
bcs_losses: [0.002445356788969817, 0.0032167091974883997, 0.0005848532314314291, 0.010004345815279583, 0.0002861292023335351, 0.01729524047288632]
loss: 0.034305732710897846
pde_losses: [0.0058697554345569555, 0.012336679874518327]
bcs_losses: [0.0004279843980850551, 0.00456837224693514, 0.00017336534123048804, 0.0017762432795465325, 0.0006908425282879814, 0.00846248960773736]
loss: 0.020801711935182012
pde_losses: [0.0046913468092250514, 0.0010192744363067606]
bcs_losses: [0.0011012324987368224, 0.005900485103007918, 0.0007862745401693177, 0.0016202137135590835, 0.00018463866453964227, 0.005498246169637417]
loss: 0.01612529410003579
pde_losses: [0.002838374928124685, 0.0014820002957597772]
bcs_losses: [0.00040154651848346506, 0.002934678441374867, 0.00020519241974271884, 0.0011248913469586441, 0.00047901208952011974, 0.006659598060071516]
loss: 0.01105632850392112
pde_losses: [0.0019045500035018994, 0.0013904169607878334]
bcs_losses: [5.7009994798387244e-5, 0.0019232243652477642, 0.00011750442212448878, 0.000626504404008349, 0.0010709078289557136, 0.003966210524496686]
loss: 0.010239250507202802
pde_losses: [0.0017479883443378445, 0.002212117084636606]
bcs_losses: [6.785173723269739e-5, 0.002189716443974777, 0.00017833917626768002, 0.0008911135335056929, 0.0009080283165650508, 0.0020440958706824538]
loss: 0.008589351810009646
pde_losses: [0.001447497915932929, 0.002378372378320729]
bcs_losses: [0.00010308237049935042, 0.0013960328915199696, 0.0002863502526423087, 0.0004816583274644143, 0.0008035292589257918, 0.0016928284147041542]
loss: 0.007174112735517012
pde_losses: [0.0008791365179751716, 0.00260527533788075]
bcs_losses: [6.758661249994968e-5, 0.0009645589515723894, 0.000379380040999735, 0.0001576780306435425, 0.0005490995680611842, 0.001571397675884289]
loss: 0.006031379152368732
pde_losses: [0.0005741393000449042, 0.0024816943729389336]
bcs_losses: [4.0879758709120445e-5, 0.0007929154784590579, 0.0004421200047049009, 0.00012554401078603168, 0.00041248305110794367, 0.0011616031756178395]
loss: 0.0048358733763691035
pde_losses: [0.00038727725155232465, 0.0022723786420140025]
bcs_losses: [2.8575675857983128e-5, 0.0005361747221107191, 0.0005318306372948938, 8.428863351035785e-5, 0.0003108453281750693, 0.0006845024858537533]
loss: 0.004014563086953044
pde_losses: [0.0004333615664716057, 0.0016710055415010549]
bcs_losses: [1.4156219369521034e-5, 0.00043599676102059834, 0.0006080530439867882, 7.292365229328564e-5, 0.00026673906598544936, 0.0005123272363247412]
loss: 0.002689580074543303
pde_losses: [0.00044489903416979157, 0.000518717309008941]
bcs_losses: [4.720057278398492e-6, 0.00029003349274222374, 0.0005019588405720959, 9.470309804256714e-5, 0.00022517828655182237, 0.000609369956177463]
loss: 0.002287946557473268
pde_losses: [0.00020739498873420574, 0.00040895904929924717]
bcs_losses: [4.463412754599384e-6, 0.00022200729967869473, 0.0004259221804879666, 0.00010923200142472106, 0.00021250319906806822, 0.0006974644260257654]
loss: 0.0021965751538584005
pde_losses: [0.0001681198814683933, 0.0004011332749079733]
bcs_losses: [1.2250570455250548e-6, 0.0002741628640461185, 0.00039008101801949044, 0.0001016577373967799, 0.0001900273106684039, 0.000670168010305716]
loss: 0.0021687830946924805
pde_losses: [0.00017638902492344536, 0.0004020469733073119]
bcs_losses: [1.3427557775527422e-6, 0.00024733817897650207, 0.0003698782900621749, 0.0001089197251204296, 0.00018546540167319958, 0.0006774027448518646]
loss: 0.0021146190245860163
pde_losses: [0.0001957362695986185, 0.00040916420681564355]
bcs_losses: [2.2876412817236343e-6, 0.00022966463307294113, 0.0003296115778454401, 0.00011964067245977069, 0.0001710507863578674, 0.0006574632371540115]
loss: 0.0020182471846700396
pde_losses: [0.00021052715971814936, 0.0004096926811260417]
bcs_losses: [1.0169263133205933e-5, 0.0002114354383982405, 0.00028049955664464176, 0.00013512659841199528, 0.0001450690391173562, 0.0006157274481204087]
loss: 0.0018555649102827872
pde_losses: [0.0001848293646639834, 0.0004128776460024795]
bcs_losses: [3.116638820621745e-5, 0.00018706134713057925, 0.00020430216818738935, 0.0001484954166462325, 0.00011532679951985344, 0.0005715057799260522]
loss: 0.0016466682333728167
pde_losses: [0.0001292333729488381, 0.00038676343243273603]
bcs_losses: [5.786223383339327e-5, 0.0001662089456096979, 0.0001320309387698816, 0.00013550406971137136, 9.652625067269356e-5, 0.0005425389893942047]
loss: 0.0014224924786299057
pde_losses: [8.627579877413215e-5, 0.0002998199609001941]
bcs_losses: [6.988586191138157e-5, 0.00016212580208636174, 9.475424742393801e-5, 0.0001093476307006292, 0.00010541984374340788, 0.0004948633330898611]
loss: 0.001312640932628529
pde_losses: [7.045561804947083e-5, 0.000262777783252638]
bcs_losses: [6.249732111020645e-5, 0.0001360512186415397, 7.830541234034736e-5, 7.40675210598312e-5, 0.00013734783930810298, 0.0004911382188663926]
loss: 0.0012864246222285963
pde_losses: [6.586958290887932e-5, 0.0002651211285984325]
bcs_losses: [4.558296399660248e-5, 0.00013308809744051037, 8.818018600481291e-5, 5.805747738716996e-5, 0.0001502586137042961, 0.0004802665721878928]
loss: 0.001238550114272252
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ps[1]

ps[2]