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:

\[\begin{aligned} u(0,x) = k * [j0(ξ(0, x)) + y0(ξ(0, x))] \\ u(t,0) = k * [j0(ξ(t, 0)) + y0(ξ(t, 0))] \\ u(t,1) = k * [j0(ξ(t, 1)) + y0(ξ(t, 1))] \\ w(0,x) = j0(ξ(0, x)) + y0(ξ(0, x)) \\ w(t,0) = j0(ξ(t, 0)) + y0(ξ(t, 0)) \\ w(t,1) = j0(ξ(t, 0)) + y0(ξ(t, 0)) \\ \end{aligned}\]

where k is a root of the algebraic (transcendental) equation f(k) = g(k), j0 and y0 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
using SpecialFunctions
using Plots
import ModelingToolkit: Interval, infimum, supremum

@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 = [Lux.Chain(Dense(input_, n, Lux.σ), Dense(n, n, Lux.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), pde_inner_loss_functions))
    println("bcs_losses: ", map(l_ -> l_(p), bcs_inner_loss_functions))
    return false
end

res = Optimization.solve(prob, BFGS(); callback = callback, maxiters=1000)

phi = discretization.phi

# Analysis
ts, xs = [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: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]
for i in 1:2
    p1 = plot(ts, xs, u_real[i], linetype=:contourf, title="u$i, analytic");
    p2 = plot(ts, xs, u_predict[i], linetype=:contourf, title="predict");
    p3 = plot(ts, xs, diff_u[i], linetype=:contourf, title="error");
    plot(p1, p2, p3)
    savefig("nonlinear_hyperbolic_sol_u$i")
end
loss: 978.9252129773065
pde_losses: [798.3005066737605, 57.860638565388186]
bcs_losses: [37.03412736628688, 0.4981005671619224, 43.56366852268714, 0.2936340531357941, 41.00259053420835, 0.37194669467775027]
loss: 459.31964707952454
pde_losses: [0.12295542377614177, 352.962285218167]
bcs_losses: [8.386963874458374, 31.088007298765376, 11.766309592359253, 21.98011521794122, 10.62336071221934, 22.389649741837882]
loss: 271.98440971562263
pde_losses: [0.8556352099963555, 67.93481493295656]
bcs_losses: [21.165504547212755, 51.67353508919941, 26.25879017546069, 39.52717557898899, 24.40429420399146, 40.16465997781641]
loss: 106.21739599592773
pde_losses: [1.1286648567247524, 0.43963100827004176]
bcs_losses: [8.872664387542544, 33.933770970976, 6.4301918372308915, 23.74432072472718, 6.695179146730265, 24.972973063726055]
loss: 103.83230789452745
pde_losses: [0.9220009263754788, 1.102652535485203]
bcs_losses: [8.071672286382453, 33.68137174266511, 5.755314166688193, 23.53452083637077, 6.006502352155553, 24.758273048404696]
loss: 25.593019728054248
pde_losses: [0.4002916589737032, 0.380505207974554]
bcs_losses: [2.332124883031912, 9.68851232721811, 1.2134019283789543, 4.8203159705973935, 1.3373963683736017, 5.420471383506018]
loss: 1.783595711236063
pde_losses: [0.009259667849498261, 0.24991282373151033]
bcs_losses: [0.14216029265748423, 0.47153199939155305, 0.06075812250737968, 0.37019992935642504, 0.0746257961258306, 0.40514707961638186]
loss: 1.5447617428845717
pde_losses: [0.02710798944135383, 0.010645678340920227]
bcs_losses: [0.10063116087271223, 0.3609137107728245, 0.0828901526951118, 0.4298630083650645, 0.08822471326028304, 0.44448532913630134]
loss: 1.519741620538565
pde_losses: [0.0013465494995437376, 0.014333400543395443]
bcs_losses: [0.10150985888543013, 0.36759497869767255, 0.08255545671365236, 0.42492473874244074, 0.08723648324354728, 0.44024015421288265]
loss: 1.5185671809820438
pde_losses: [0.0011957227327097201, 0.013013625995875694]
bcs_losses: [0.1032354449544071, 0.36781206087956153, 0.0818913108423405, 0.4246186679141414, 0.08700303586490299, 0.4397973117981046]
loss: 1.5139947398964069
pde_losses: [0.002317654614784321, 0.004724015446007789]
bcs_losses: [0.10564473053265265, 0.3662796623832054, 0.08243405720265762, 0.42463046588988285, 0.08942290795422478, 0.4385412458729917]
loss: 1.5131824794087438
pde_losses: [0.0022205476271353686, 0.004690136710899354]
bcs_losses: [0.10472442124449034, 0.36673470118767093, 0.0829997374420467, 0.42408667893971713, 0.08986881954181553, 0.4378574367149686]
loss: 1.5115138592601152
pde_losses: [0.002516374174176184, 0.0023758817005613045]
bcs_losses: [0.10641736667383146, 0.3644946572922025, 0.08315474254716292, 0.4247429295059135, 0.0905203240609899, 0.43729158330527723]
loss: 1.5089383656131916
pde_losses: [0.0012561003369892512, 0.0024310708878796314]
bcs_losses: [0.1057101367705291, 0.36317660249973277, 0.08424995345466625, 0.42489218613635005, 0.09074183919513606, 0.4364804763319084]
loss: 1.5080422183248485
pde_losses: [0.000956716381962312, 0.0025261452189195106]
bcs_losses: [0.10648468147896911, 0.3657326411973718, 0.08391266937507472, 0.42308461364140315, 0.09033009018316944, 0.4350146608479786]
loss: 1.507224909999371
pde_losses: [0.000995167282573717, 0.0037768640861998094]
bcs_losses: [0.10496261302865816, 0.3623547981452453, 0.0838018179162174, 0.42514098616000984, 0.08994290698396798, 0.4362497563964988]
loss: 1.5061921049079927
pde_losses: [0.0011433313246351031, 0.0063556859149807]
bcs_losses: [0.10454874305135965, 0.3662788103176407, 0.08208639794541886, 0.4225406171330255, 0.08889198630612255, 0.43434653291480946]
loss: 1.5059079509757969
pde_losses: [0.0012211591463629175, 0.005848426951374484]
bcs_losses: [0.10378163167901243, 0.36346789187715317, 0.08228379999262818, 0.42430864708236726, 0.0895062490871432, 0.43549014515975515]
loss: 1.505846564270491
pde_losses: [0.0011630469061655103, 0.005476391995221798]
bcs_losses: [0.10440338693473081, 0.3655887707271905, 0.08226422772766791, 0.4229392061784606, 0.0894738664804275, 0.4345376673206264]
loss: 1.5058030257005002
pde_losses: [0.0011931113589501296, 0.005137414230989502]
bcs_losses: [0.10477682757179123, 0.3658517351168251, 0.08219056364292898, 0.4227610268246244, 0.08950292464094346, 0.4343894223134469]
loss: 1.5057712107508532
pde_losses: [0.0012265953932720512, 0.005568076197583894]
bcs_losses: [0.10415041739883867, 0.3647163976709334, 0.08226914077266864, 0.4234830108718226, 0.08949212644462032, 0.4348654460011136]
loss: 1.5056332072390266
pde_losses: [0.0011202922172340926, 0.005678092214012234]
bcs_losses: [0.10411026659856515, 0.3636047327374099, 0.08223417844415427, 0.4241939668739519, 0.08936254460323267, 0.43532913355046654]
loss: 1.5051967952717302
pde_losses: [0.0008794113671056806, 0.005816328828430223]
bcs_losses: [0.10451765337678445, 0.36554445338921365, 0.08186717858477269, 0.42295193242962614, 0.0891556880824544, 0.43446414921334287]
loss: 1.5049242381471029
pde_losses: [0.0007378329624843095, 0.006156574889848436]
bcs_losses: [0.10463754675426738, 0.36661934158378195, 0.08159450815741465, 0.42226813423665244, 0.0889154296783703, 0.4339948698842835]
loss: 1.5047038644963016
pde_losses: [0.0005490942948984882, 0.006546320816344002]
bcs_losses: [0.10416725833575236, 0.366931152903463, 0.08180513988308853, 0.422067279806703, 0.08877468141144712, 0.43386293704460505]
loss: 1.5046788399672986
pde_losses: [0.0005866319974857034, 0.00637233553272682]
bcs_losses: [0.1037440552827621, 0.36529179810537776, 0.08204065813955902, 0.4231137076109386, 0.08895654037339945, 0.4345731129250491]
loss: 1.5046766144216868
pde_losses: [0.0005853866397431931, 0.006545519214391746]
bcs_losses: [0.10366203086473234, 0.365243770184116, 0.08198061178721752, 0.423144413877569, 0.08892083007206149, 0.4345940517818556]
loss: 1.5046722419752352
pde_losses: [0.0005783689029870291, 0.00660920606366591]
bcs_losses: [0.10362423453070893, 0.3654997448372464, 0.08196542385229331, 0.4229806709964821, 0.08893198010738534, 0.4344826126844662]
loss: 1.5046222069260362
pde_losses: [0.0005882834568882446, 0.006341406541254872]
bcs_losses: [0.10415096236418103, 0.36713628787161157, 0.08181252230414264, 0.42193810442401425, 0.08888244870209902, 0.4337721912618445]
loss: 1.50451723319379
pde_losses: [0.0005645060043703241, 0.006167399405027433]
bcs_losses: [0.1045025044966577, 0.3679424435256588, 0.0816512784164417, 0.42142794620501917, 0.08883788620167499, 0.4334232689389399]
loss: 1.5043961189424857
pde_losses: [0.0004844079507776057, 0.005920349046803511]
bcs_losses: [0.10469716595937924, 0.3680384657920519, 0.08160982475213188, 0.4213684864522952, 0.08889625797102174, 0.43338116101802443]
loss: 1.5042272423060985
pde_losses: [0.0003848214581899548, 0.005640891104718336]
bcs_losses: [0.10419395997167212, 0.366897363511554, 0.08193590094873957, 0.42209469171582137, 0.0892070232524471, 0.4338725903429561]
loss: 1.5041727986926132
pde_losses: [0.0003674914140894359, 0.005961967182688733]
bcs_losses: [0.10393155894265985, 0.36603806559489716, 0.08190299301474807, 0.42264238020686706, 0.08908362512481897, 0.434244717211844]
loss: 1.5041090263074863
pde_losses: [0.0004135750887239397, 0.006143462722302258]
bcs_losses: [0.10319672671709354, 0.3641988257482326, 0.08205897111913407, 0.4238211113044455, 0.08923016742766465, 0.4350461861798897]
loss: 1.5040310135213106
pde_losses: [0.0004243157746396517, 0.006107154685731885]
bcs_losses: [0.10315150692029823, 0.3647290261626333, 0.08212262353606263, 0.42348164054026927, 0.0892003454137715, 0.4348144004879042]
loss: 1.5040070893196757
pde_losses: [0.0004367157314776291, 0.0061503954667818635]
bcs_losses: [0.103227110208248, 0.3651243651633722, 0.08203334386095396, 0.42322860191116085, 0.08916459973565707, 0.4346419572420242]
loss: 1.50396655569569
pde_losses: [0.000454224373646847, 0.006843713719777127]
bcs_losses: [0.1031066420451103, 0.36667304143207713, 0.08180928476356232, 0.4222402606761795, 0.08886992108118676, 0.43396946760415017]
loss: 1.503956361400403
pde_losses: [0.00047294596744781376, 0.006658815698811351]
bcs_losses: [0.10311490543275718, 0.3665033444565953, 0.08184728025949335, 0.4223484517862636, 0.08896768580189741, 0.4340429319971367]
loss: 1.5039479861873448
pde_losses: [0.00047504756735935647, 0.006474309463791637]
bcs_losses: [0.10313802544149886, 0.3659865812353537, 0.08190145123128648, 0.4226779890113615, 0.08902760551050196, 0.4342669767261913]
loss: 1.5039347307431759
pde_losses: [0.0004929929367391747, 0.006383251026696826]
bcs_losses: [0.10282240608461772, 0.364994713465969, 0.08204859901185446, 0.42331225083482765, 0.08918200246696524, 0.4346985149155058]
loss: 1.5039006754205266
pde_losses: [0.0005105729714483917, 0.006623165226767682]
bcs_losses: [0.10250516471292348, 0.3642735223432025, 0.08200273395875976, 0.423775017860055, 0.08919695216045584, 0.435013546186914]
loss: 1.5038138664169254
pde_losses: [0.0005007916271985712, 0.006702680156767348]
bcs_losses: [0.1023009975464747, 0.36368792763192354, 0.08198749631726861, 0.42415180362010446, 0.08921203443486186, 0.4352701350823262]
loss: 1.5036609587448262
pde_losses: [0.00048641178151223466, 0.0069712522308607045]
bcs_losses: [0.10198440074124669, 0.363643001476716, 0.0820841027171018, 0.4241809325599949, 0.0890209862944829, 0.435289870942911]
loss: 1.5035695725622835
pde_losses: [0.0004683161185828234, 0.006540837063322219]
bcs_losses: [0.10223978173297957, 0.3652737526394259, 0.08223357778171671, 0.42313449003502435, 0.08910181138990242, 0.43457700580132963]
loss: 1.5034594548107603
pde_losses: [0.0004332133715261127, 0.0070051319782398055]
bcs_losses: [0.10270796032433346, 0.3678629297667646, 0.08185039052002147, 0.42148546424632816, 0.08865820740406485, 0.4334561571994817]
loss: 1.5034238382031622
pde_losses: [0.0004234338645769363, 0.007575265601913349]
bcs_losses: [0.10234639029041478, 0.3681886691299582, 0.08183733888436552, 0.4212790990677836, 0.08845754328161946, 0.43331609808253024]
loss: 1.5033979626782135
pde_losses: [0.00042060698895611336, 0.007626860759676533]
bcs_losses: [0.10229054564613387, 0.3680635785455768, 0.08184705730521968, 0.4213583723864728, 0.08842107050749536, 0.43336987053868237]
loss: 1.5033661188587901
pde_losses: [0.0004201120427618183, 0.007629679480726229]
bcs_losses: [0.10218828371565397, 0.3677543058957336, 0.08190875947249086, 0.42155446831660265, 0.08840756952230927, 0.43350294041251175]
loss: 1.503324341884175
pde_losses: [0.00042976957360082963, 0.007590176193748132]
bcs_losses: [0.10196150850383963, 0.3669172762025628, 0.08206804112588316, 0.4220862260394624, 0.0884073114466192, 0.4338640327984589]
loss: 1.5032977791629516
pde_losses: [0.0004492109748774974, 0.007479339243980673]
bcs_losses: [0.10145976562754841, 0.36523455337387156, 0.08239538399033193, 0.4231599949192396, 0.08852535338184045, 0.4345941776512614]
loss: 1.503286639046994
pde_losses: [0.00046786900223182404, 0.0076348910217259236]
bcs_losses: [0.10147636564179477, 0.3656226509740066, 0.08232083402463451, 0.42291180881245105, 0.08842693136169527, 0.43442528820845405]
loss: 1.5032820608480928
pde_losses: [0.00046761880574645013, 0.007657331432861716]
bcs_losses: [0.10147576710474773, 0.3659799460666373, 0.08231080611440844, 0.42268360233317115, 0.08843692628235801, 0.434270062708162]
loss: 1.5032788536224064
pde_losses: [0.00048267298197053724, 0.007663468843500513]
bcs_losses: [0.10160161769998932, 0.36644382803633263, 0.08225243705795424, 0.42238778036692215, 0.08837812611968202, 0.43406892251605494]
loss: 1.5032767763990034
pde_losses: [0.0004918396839095369, 0.00752048969415449]
bcs_losses: [0.10165107230738914, 0.36642543775504066, 0.08230863962371265, 0.42239950835316126, 0.08840289723993966, 0.4340768917416959]
loss: 1.5032576492085277
pde_losses: [0.0004945198330892556, 0.00751220473532071]
bcs_losses: [0.10128914071529566, 0.3651197157783507, 0.08244384426524037, 0.42323356208460106, 0.08852043141381499, 0.43464423038281513]
loss: 1.503193018280452
pde_losses: [0.00042734627010015475, 0.007686933380024398]
bcs_losses: [0.10096786879033351, 0.3644508764638493, 0.08244357440479014, 0.4236622509319952, 0.08861802156912676, 0.4349361464702327]
loss: 1.5031594695836088
pde_losses: [0.000382158964485927, 0.00781033529706291]
bcs_losses: [0.10099742726559782, 0.3646991057855152, 0.08236539693527468, 0.42350296183827274, 0.0885744095423886, 0.43482767395501093]
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loss: 5.385142866864912e-5
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bcs_losses: [3.475202521973445e-6, 3.1728016752155413e-6, 1.574011202936939e-6, 3.4744808258814595e-6, 7.5743120133326204e-6, 1.0215547026158264e-5]
loss: 5.3560910018676485e-5
pde_losses: [8.166326375179159e-6, 1.5880101962072852e-5]
bcs_losses: [3.439828198982179e-6, 3.163494318077318e-6, 1.5582778093277398e-6, 3.653591710262001e-6, 7.543354378880059e-6, 1.0155935265895178e-5]
loss: 5.3096120903658024e-5
pde_losses: [7.967424321916418e-6, 1.5649526811588502e-5]
bcs_losses: [3.4210755929888505e-6, 3.248910617250253e-6, 1.3355938170065326e-6, 3.957588328888575e-6, 7.494657985551106e-6, 1.002134342846778e-5]

nonlinear_hyperbolic_sol_u1 nonlinear_hyperbolic_sol_u2