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 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 = [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 = [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]
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: 9.791452874758956
pde_losses: [0.01749455777752525, 5.399580317208114]
bcs_losses: [0.020016309441497418, 0.9933823610548266, 0.05844085750157908, 1.6715958482903859, 0.07392026039044299, 1.557022363094584]
loss: 6.155403156897311
pde_losses: [0.016841832663023394, 2.349524389258951]
bcs_losses: [0.005798953606792673, 0.8503419059742352, 0.03228893736583357, 1.4842507435527377, 0.042892021626414305, 1.3734643728493234]
loss: 3.2780350440556862
pde_losses: [0.008125748121976608, 0.023264096527663734]
bcs_losses: [0.0007006177705300508, 0.7178872161656626, 0.007714773415812395, 1.3074626807980818, 0.012112907558466755, 1.2007670036974925]
loss: 1.2001466423239129
pde_losses: [0.020534992982711973, 0.1988157239033404]
bcs_losses: [0.12321194730949911, 0.02993387742919324, 0.19809540509585613, 0.22406245571933808, 0.23326359288370432, 0.17222864700026963]
loss: 0.8450971934103889
pde_losses: [0.008018551698120916, 0.010721196216541079]
bcs_losses: [0.13020792217906296, 0.005086000203184033, 0.21704852459841412, 0.13858093339909722, 0.24303273298159606, 0.09240133213437247]
loss: 0.7894427076481297
pde_losses: [0.0018262057511936735, 0.020883651562671187]
bcs_losses: [0.12106433329084677, 0.002817992241637309, 0.21552040103874864, 0.12204580706890318, 0.23209870624076584, 0.07318561045336316]
loss: 0.70329740663712
pde_losses: [0.004543019105520457, 0.02065285741276238]
bcs_losses: [0.09428296559096698, 0.0039708338215575616, 0.20435577000326838, 0.12683555004522248, 0.19514854236644336, 0.05350786829137844]
loss: 0.6615999839211885
pde_losses: [0.006288675374044461, 0.03602864274436619]
bcs_losses: [0.08142526098951355, 0.004699155574426518, 0.19119286307932132, 0.12324735399169176, 0.176220261830823, 0.04249777033700167]
loss: 0.5972374410913085
pde_losses: [0.012131644785028992, 0.1198268456902992]
bcs_losses: [0.04977015865539494, 0.009955070775917101, 0.15467495440472306, 0.1023730862486183, 0.12786324721935305, 0.020642433311973837]
loss: 0.5419598756380989
pde_losses: [0.010362565426816524, 0.12335812772965349]
bcs_losses: [0.04386868159422512, 0.022921206887053056, 0.14864740729409115, 0.062004179374322385, 0.11610418146077685, 0.014693525871160334]
loss: 0.47357053851846165
pde_losses: [0.026630845726486156, 0.2308650349920594]
bcs_losses: [0.005990097940157187, 0.02643874468252544, 0.07169447524780935, 0.057034201500523955, 0.03932236326985441, 0.015594775159045697]
loss: 0.30908371570615856
pde_losses: [0.031423879902701506, 0.0815553128186351]
bcs_losses: [0.004617863511913177, 0.026648264029381766, 0.0629980803118185, 0.05169911417552015, 0.034665369555413834, 0.015475831400774496]
loss: 0.2608985983865489
pde_losses: [0.027626411181797015, 0.05624915426612139]
bcs_losses: [0.002699352835668937, 0.034730229911301175, 0.05318066635471799, 0.03809568194742698, 0.02856030153821458, 0.01975680035130079]
loss: 0.11488486780193391
pde_losses: [0.003144194548715315, 0.004174708389624271]
bcs_losses: [0.00026674441846462, 0.00867317894452454, 0.017062898796598478, 0.05103519746069238, 0.012588530138328557, 0.01793941510498576]
loss: 0.0849342852837878
pde_losses: [0.003075790831761335, 0.004636569425465815]
bcs_losses: [0.0023087641441368045, 0.01915753430491976, 0.009392879661199786, 0.026387013818969934, 0.006551973089989861, 0.013423760007344503]
loss: 0.07944968383635376
pde_losses: [0.003265883705389232, 0.003045638484398854]
bcs_losses: [0.0039046027911863786, 0.022074153624190247, 0.007184101860342141, 0.022041260949352173, 0.005051368313871031, 0.012882674107623693]
loss: 0.06505722145322348
pde_losses: [0.0008237877692634631, 0.0020235387789987957]
bcs_losses: [0.007320259332246091, 0.02417868771641909, 0.004370162961680282, 0.012098362278653418, 0.003504886831673973, 0.010737535784288378]
loss: 0.05408728764215051
pde_losses: [0.0008984556490566544, 0.004374031673166975]
bcs_losses: [0.008299609153537228, 0.01502417035489275, 0.0035066096680400416, 0.009394680603544824, 0.0031507739440600683, 0.00943895659585197]
loss: 0.04725242410352883
pde_losses: [0.0018308897481288007, 0.00940406361710879]
bcs_losses: [0.00811637633261218, 0.0071890393655487115, 0.0026061705593897334, 0.005488370052358112, 0.0027119921723585926, 0.009905522256023917]
loss: 0.04405516084144648
pde_losses: [0.002173718045298583, 0.013484027474484664]
bcs_losses: [0.00891522752995695, 0.0014894242465349151, 0.0014123345873346953, 0.001909218381573177, 0.002259584007343849, 0.012411626568919639]
loss: 0.03244276508876802
pde_losses: [0.0013639198554026277, 0.006128084842865706]
bcs_losses: [0.00796237432350527, 0.0037087535837093144, 0.0013508781735227361, 0.0011026583485586518, 0.002069493937929235, 0.008756602023274475]
loss: 0.026029551667031667
pde_losses: [0.0013201621746017592, 0.0051874935906370805]
bcs_losses: [0.006122055810521705, 0.0028060436274213642, 0.0012720580646686688, 0.0007378602307263616, 0.0017831256536992059, 0.006800752514755523]
loss: 0.011898056723465718
pde_losses: [0.0010350504609482098, 0.005817138997384451]
bcs_losses: [0.0023271186353744366, 0.0008137876162066832, 0.00026778307895193414, 0.0004448800992207659, 0.0007240222516159626, 0.00046827558376327566]
loss: 0.007860214044779
pde_losses: [0.0011449223969272919, 0.0025200650104107644]
bcs_losses: [0.0016658329592991032, 0.0006391103965211403, 0.00018014404257508902, 0.0007538419077728992, 0.0006324420465928362, 0.0003238552846798757]
loss: 0.005717669487559116
pde_losses: [0.0012389715519713744, 0.001492727674003633]
bcs_losses: [0.0007571368460994783, 0.0002715987783408901, 0.0001816771150581395, 0.0006609856127621818, 0.0007802324685334549, 0.0003343394407899642]
loss: 0.005288783242935247
pde_losses: [0.0010842439272688487, 0.001163446168132442]
bcs_losses: [0.0006431941396705337, 0.00020612766052480254, 0.0002262655723169272, 0.0007510194958943991, 0.0008521061083425042, 0.00036238017078478853]
loss: 0.0048252099567477
pde_losses: [0.0009458374388595468, 0.0009490082222336546]
bcs_losses: [0.000626293354883513, 0.00021300981067992527, 0.00022153540365505022, 0.0007090651150423689, 0.0007977612683848729, 0.00036269934300876773]
loss: 0.004100736669862881
pde_losses: [0.0007368058027366313, 0.0008187012283005642]
bcs_losses: [0.0006250205058289735, 0.00023766623296330938, 9.963337587349358e-5, 0.0007261516911377738, 0.0005075382590076056, 0.00034921957401453047]
loss: 0.0036479198833353783
pde_losses: [0.0004337317305550902, 0.0008638519172055811]
bcs_losses: [0.000700202314008966, 0.00021159938919861206, 8.383324436206202e-5, 0.0006370159140965498, 0.0003762868745019559, 0.000341398499406561]
loss: 0.0027841078297440964
pde_losses: [0.00021788451849954932, 0.00039263044300235375]
bcs_losses: [0.000574743923751733, 0.00015661368988616753, 7.580485095638151e-5, 0.0006958409005599705, 0.00030539066624450895, 0.00036519883684343164]
loss: 0.002282167489467411
pde_losses: [0.00010430911093690674, 0.00024961085166762617]
bcs_losses: [0.0004136356092609931, 0.00012915065455122154, 8.327769960411627e-5, 0.0006954946167579099, 0.00022706903831505617, 0.0003796199083735814]
loss: 0.002210379418281526
pde_losses: [0.00011209520723414052, 0.0001937974327891535]
bcs_losses: [0.0004351777852348737, 0.00012528430816410207, 9.743097438281926e-5, 0.0006397694520090977, 0.00022739064671739177, 0.00037943361174994745]
loss: 0.002151620489742805
pde_losses: [0.000140018876171398, 0.0001718349206519206]
bcs_losses: [0.0004156544754397539, 0.0001358736976710538, 0.00010002625620150067, 0.0006080597446927714, 0.00021402765978504012, 0.00036612485912936607]
loss: 0.002077140769392702
pde_losses: [0.000173307636539187, 0.00015531523703629876]
bcs_losses: [0.0003842207832399489, 0.00015336709032329334, 0.00010045280803516266, 0.0005590170610595713, 0.00020258940245484493, 0.00034887075070439517]
loss: 0.002029494280703959
pde_losses: [0.00022389733920191756, 0.00016138733391363302]
bcs_losses: [0.00035967464259823763, 0.00015850265425246742, 8.77089224433863e-5, 0.0005186174689895539, 0.0001750129544815766, 0.00034469296482318626]
loss: 0.0019566184070174673
pde_losses: [0.00020394496007647038, 0.00015354052835094914]
bcs_losses: [0.00033704105849909657, 0.00014828967636842583, 8.298525196530462e-5, 0.0004884381883474869, 0.00017986316223357496, 0.00036251558117615906]
loss: 0.0018759084916093003
pde_losses: [0.0001760508611208305, 0.00015507494798144605]
bcs_losses: [0.00028538578985142296, 0.00011599358166417578, 7.556594982171294e-5, 0.0004633558527366664, 0.0001799812747666922, 0.0004245002336663537]
loss: 0.0018313152436726736
pde_losses: [0.00013185519193840617, 0.00015308393122464683]
bcs_losses: [0.0002593940139780204, 9.026039082702024e-5, 6.961233484650793e-5, 0.00045351790897074174, 0.00018551564859826004, 0.0004880758232890703]
loss: 0.0017954416524476758
pde_losses: [0.00012844832868034897, 0.0001576638281937901]
bcs_losses: [0.00021348758911288824, 8.659021806556263e-5, 6.752975953144285e-5, 0.00044599259128149835, 0.00017355195903206834, 0.0005221773785500766]
loss: 0.0017349304707335667
pde_losses: [0.00010137850439004197, 0.00015354141700057612]
bcs_losses: [0.0001969224181165686, 8.041631740284081e-5, 6.684178151024826e-5, 0.00043804947302137026, 0.00016823217347206856, 0.0005295483858198522]
loss: 0.0015816329190082015
pde_losses: [6.839747115099344e-5, 0.00019392069728769464]
bcs_losses: [0.00015151383681771276, 8.503497168400336e-5, 7.095749054838096e-5, 0.00039627976997179474, 0.0001463661945440969, 0.0004691624870035247]
loss: 0.00152321077651185
pde_losses: [7.499989933677907e-5, 0.00020463067671883963]
bcs_losses: [0.00015759445360858672, 0.00010587116547380912, 7.392990835338625e-5, 0.0003785674200173955, 0.00012884883178734032, 0.0003987684212157132]
loss: 0.0015053092583422857
pde_losses: [8.673502512413025e-5, 0.0002302952347935829]
bcs_losses: [0.00014460400416043328, 0.00013144941218797902, 7.900529983213481e-5, 0.00036387925013311296, 0.00011746027127103355, 0.0003518807608398792]
loss: 0.00149132799038722
pde_losses: [7.698377567807832e-5, 0.00022238624547240462]
bcs_losses: [0.00015127320808986595, 0.00013938252983981982, 8.112066707192597e-5, 0.00037383085490085347, 0.00011264278822413802, 0.00033370792111013384]
loss: 0.0014820970895737657
pde_losses: [5.7418014391483974e-5, 0.00019561023644503256]
bcs_losses: [0.00016450312738013054, 0.00013668330948575315, 8.158321924922325e-5, 0.000392008934854277, 0.00011940809529238945, 0.0003348821524754759]
loss: 0.0014577841798233705
pde_losses: [5.085457486172531e-5, 0.00018750871473752963]
bcs_losses: [0.00015481393100870574, 0.00013380111905709336, 8.231456825119593e-5, 0.00038969370678617764, 0.00012032870333000229, 0.00033846886179094083]
loss: 0.0013696283002775013
pde_losses: [2.24094568506358e-5, 0.0001768489671329469]
bcs_losses: [0.00011638146294465574, 0.00013952217944276932, 8.62604114446418e-5, 0.00037375075774810965, 0.00011941120758712366, 0.00033504385712661836]
loss: 0.0013171912901892315
pde_losses: [1.383117700047954e-5, 0.00015503127649179943]
bcs_losses: [9.755487153162686e-5, 0.00014286335904566236, 8.894980569099125e-5, 0.00036692284177075135, 0.00012061399887176754, 0.0003314239597861531]
loss: 0.0012586789818287698
pde_losses: [2.1858932649451415e-5, 0.0001164894220385761]
bcs_losses: [8.072028278476532e-5, 0.00015070213920746026, 9.156958863029356e-5, 0.00035538991453094745, 0.00011812479202684731, 0.0003238239099604284]
loss: 0.0012268386677239407
pde_losses: [2.918049919024588e-5, 0.00010462866650884198]
bcs_losses: [7.152381218683211e-5, 0.0001568415298027034, 9.276462346435056e-5, 0.0003417919274072594, 0.00011351314538742422, 0.00031659446377628307]
loss: 0.0011939775194037712
pde_losses: [3.748830512304223e-5, 0.00010369464920352117]
bcs_losses: [6.152086089964918e-5, 0.00016305863553344696, 9.330986208229997e-5, 0.0003218263965686742, 0.00010653746996797254, 0.00030654134002516494]
loss: 0.0011536924592363278
pde_losses: [4.9103158589741654e-5, 0.00010669156881166057]
bcs_losses: [5.023285161112881e-5, 0.0001715882364412921, 9.228267961097334e-5, 0.00029586405504710605, 9.652733804740815e-5, 0.00029140257107701707]
loss: 0.001113546404603112
pde_losses: [7.59926041179368e-5, 0.00012940284410751878]
bcs_losses: [4.282790383692205e-5, 0.00016187446299489592, 8.509634672555608e-5, 0.0002730261954361559, 7.582069502146103e-5, 0.00026950535236266554]
loss: 0.0010811231263775593
pde_losses: [9.540227796189535e-5, 0.00011532569078152397]
bcs_losses: [3.886421000545592e-5, 0.00018823465381724168, 8.207050931608284e-5, 0.0002533261760809575, 6.652981441710546e-5, 0.00024136979399729658]
loss: 0.0010379206194593318
pde_losses: [8.523484654640759e-5, 9.90803095138597e-5]
bcs_losses: [3.4586923168089476e-5, 0.00016391773510773675, 7.921734233312947e-5, 0.00024240603518079013, 7.258402154995222e-5, 0.0002608934060593663]
loss: 0.0010174984544313395
pde_losses: [6.263846821950526e-5, 8.142037949332883e-5]
bcs_losses: [3.435372132977644e-5, 0.0001238159901357561, 8.358900542173175e-5, 0.0002506515207487359, 7.750024923069531e-5, 0.00030352911985180993]
loss: 0.0009663339385303441
pde_losses: [7.62344962018269e-5, 7.029983411758921e-5]
bcs_losses: [3.411712102917442e-5, 0.00014260720712802613, 8.075968219861272e-5, 0.00023680852984811165, 6.341191741673921e-5, 0.00026209515059026384]
loss: 0.0009465812497279623
pde_losses: [8.981267066999686e-5, 7.569355858607074e-5]
bcs_losses: [3.5402551669944445e-5, 0.00014997616950965118, 7.726965232060493e-5, 0.00022800204666610278, 5.546471798161979e-5, 0.00023495988232397156]
loss: 0.000912408951141683
pde_losses: [8.509239986561164e-5, 4.5844996060057206e-5]
bcs_losses: [3.458950308721784e-5, 0.00012162558188923535, 7.667448030796402e-5, 0.0002353867203385244, 5.7621895947970825e-5, 0.00025557337364510166]
loss: 0.000892421380876997
pde_losses: [9.37796603960441e-5, 3.518686037290168e-5]
bcs_losses: [3.288699935473913e-5, 0.00010440438596026541, 7.51344756945887e-5, 0.00023156565919039748, 5.702211446926824e-5, 0.00026244122543879223]
loss: 0.0008814186114645793
pde_losses: [0.0001123353685974204, 2.8148974571910818e-5]
bcs_losses: [3.368100450073443e-5, 0.00010528770613275, 7.411333182362461e-5, 0.0002234964293606218, 5.40396775206375e-5, 0.0002503161189568797]
loss: 0.0008726056033939142
pde_losses: [0.00012302566501695274, 2.9317591859199143e-5]
bcs_losses: [3.322553356015195e-5, 0.00010409149333466579, 7.281066884031628e-5, 0.00021579309345265548, 5.2158664221324865e-5, 0.00024218289310864793]
loss: 0.000853194216342638
pde_losses: [0.0001394623726726307, 2.8677559042807926e-5]
bcs_losses: [3.0180039325248575e-5, 0.00010115460500156553, 7.151786468039791e-5, 0.00019739759748125, 4.982383155647128e-5, 0.00023498034658226604]
loss: 0.0008436576726440014
pde_losses: [0.00014572797058024853, 3.464747068304312e-5]
bcs_losses: [2.9832890963739565e-5, 0.00010502667992404435, 7.040265949057158e-5, 0.00018619596384522328, 4.86883139177958e-5, 0.00022313572323933524]
loss: 0.0008123277102126314
pde_losses: [0.0001428168058334737, 6.135773779395052e-5]
bcs_losses: [2.5181340479725324e-5, 0.0001218537234381807, 6.834073719187506e-5, 0.00014641125669872783, 4.630414430342861e-5, 0.0002000619644732696]
loss: 0.0007874448611137875
pde_losses: [0.0001331522764775074, 5.227033369458019e-5]
bcs_losses: [2.2533335752725382e-5, 0.00012023916991016612, 6.899428772613034e-5, 0.00013651311668458333, 4.5994306392686884e-5, 0.00020774803447540794]
loss: 0.0007636617708062735
pde_losses: [0.00013429419594543457, 4.866249910672516e-5]
bcs_losses: [1.743617770632004e-5, 0.00010463678495673617, 7.304431162547027e-5, 0.00012311977134488376, 4.837816903898539e-5, 0.00021408986108171815]
loss: 0.0007508401621614511
pde_losses: [0.00013741412072604576, 6.936129850593925e-5]
bcs_losses: [1.249955388935553e-5, 0.00011272890093481085, 7.314332842709824e-5, 9.978641528101629e-5, 4.952716452220499e-5, 0.00019637937987498016]
loss: 0.0007378821880242687
pde_losses: [0.0001346617257909037, 6.0738099555216495e-5]
bcs_losses: [9.846221575613399e-6, 9.506210845827564e-5, 7.520212895378418e-5, 9.736394558109557e-5, 5.2782322786837234e-5, 0.00021222563532254256]
loss: 0.0007158043323857288
pde_losses: [0.00013491686928754405, 4.6089186084874755e-5]
bcs_losses: [1.013276247995967e-5, 9.550867651002078e-5, 7.783354310911458e-5, 9.320255115666167e-5, 5.291387605277147e-5, 0.00020520686770478176]
loss: 0.0007088478732286026
pde_losses: [0.00013598487136917814, 6.188363902238267e-5]
bcs_losses: [1.0024582113402266e-5, 9.35408190769972e-5, 8.056722652815481e-5, 8.426908743597052e-5, 5.5519143681905425e-5, 0.0001870585040006115]
loss: 0.0006914104838376109
pde_losses: [0.00013386461797312743, 4.618097027372854e-5]
bcs_losses: [1.024029621540071e-5, 8.573081289479371e-5, 8.168997285851731e-5, 8.719951468364828e-5, 5.554990150629761e-5, 0.00019095439743209728]
loss: 0.0006693667960444112
pde_losses: [0.00013127782453506233, 4.751231118596952e-5]
bcs_losses: [1.1230533522977199e-5, 8.396912349807632e-5, 8.44542287780811e-5, 8.086148523643892e-5, 5.750780059059301e-5, 0.00017255348869721267]
loss: 0.0006394042064598771
pde_losses: [0.00012457487870844053, 5.645499355557314e-5]
bcs_losses: [2.1971449339614884e-5, 7.51254104944163e-5, 8.872353333846467e-5, 7.256796331923047e-5, 6.521015550294932e-5, 0.00013477582220118773]
loss: 0.0006362954129288821
pde_losses: [0.00012338425974736507, 5.8995841105421925e-5]
bcs_losses: [1.6853172934699468e-5, 7.22201811295334e-5, 8.42511109020059e-5, 7.509242187202816e-5, 6.232165944153126e-5, 0.00014317676579629694]
loss: 0.0006270598955413403
pde_losses: [0.00012252976385089927, 5.616379845641012e-5]
bcs_losses: [1.584667400248471e-5, 8.635950792017667e-5, 8.1031946140971e-5, 6.995406318723019e-5, 6.222403824019525e-5, 0.00013295010374297318]
loss: 0.0006190082587573173
pde_losses: [0.00011815312052497041, 5.7988241516453275e-5]
bcs_losses: [1.691600952496008e-5, 8.809273607199318e-5, 8.163311306754081e-5, 6.665297572759708e-5, 6.287154195237645e-5, 0.00012670052037142593]
loss: 0.0006116143632404461
pde_losses: [0.00011603128421122544, 6.0142160325253556e-5]
bcs_losses: [1.6355007061395182e-5, 8.525312605386489e-5, 7.91589146804099e-5, 6.735322560833586e-5, 6.165621503002934e-5, 0.00012566443026993196]
loss: 0.0005956644528335328
pde_losses: [0.00011132001732971114, 6.895058376430227e-5]
bcs_losses: [1.2042532280583898e-5, 9.330786979629558e-5, 7.264235127834453e-5, 6.206415175654088e-5, 5.690791479836193e-5, 0.00011842903182939264]
loss: 0.0005798748652656534
pde_losses: [0.00010605045625424962, 7.520689039791421e-5]
bcs_losses: [9.770260968677376e-6, 9.158330622735429e-5, 6.523891022268244e-5, 6.260780724725916e-5, 5.183621411858331e-5, 0.00011758101982893291]
loss: 0.0005627799076434549
pde_losses: [0.00010669061677617956, 6.792922920344074e-5]
bcs_losses: [9.646747521675773e-6, 9.64615483518983e-5, 6.101240976333108e-5, 5.531214670008249e-5, 4.604654755465336e-5, 0.00011968066177219362]
loss: 0.0005434710690964096
pde_losses: [0.00011063988241281808, 5.87988740589941e-5]
bcs_losses: [9.423757028472052e-6, 8.499049065175844e-5, 6.15818076898788e-5, 5.441351261732533e-5, 4.500351279255502e-5, 0.00011861923184460773]
loss: 0.0005193853698285724
pde_losses: [0.00012759855006823587, 4.487931699785319e-5]
bcs_losses: [1.0011087061078225e-5, 7.750151657159201e-5, 6.520604230004503e-5, 4.58450357517779e-5, 4.4787066917709245e-5, 0.000103556754160281]
loss: 0.0005117804914977199
pde_losses: [0.00013853462316671563, 4.137616223552225e-5]
bcs_losses: [1.0667187986954162e-5, 6.949527178132391e-5, 6.45704978016259e-5, 4.42407097589173e-5, 4.339400582872716e-5, 9.950203293793357e-5]
loss: 0.0005024199865565312
pde_losses: [0.0001501050100383833, 4.528798172256562e-5]
bcs_losses: [1.2395071916207431e-5, 5.380587663057445e-5, 6.444083947481365e-5, 4.2892068220367013e-5, 4.280917866380301e-5, 9.068395988981664e-5]
loss: 0.0004828382421704909
pde_losses: [0.00016679369577346306, 3.805854317322568e-5]
bcs_losses: [1.3934477727220129e-5, 4.24104374069691e-5, 6.305205794618591e-5, 3.638196099179077e-5, 3.8676420522970965e-5, 8.35306486286653e-5]
loss: 0.0004488458657325085
pde_losses: [0.00017504882899198448, 3.468913573833574e-5]
bcs_losses: [1.9374335243656455e-5, 2.9192292617981896e-5, 5.857755448059631e-5, 3.055040322649632e-5, 3.292888264086592e-5, 6.84844327925913e-5]
loss: 0.0004263268974827836
pde_losses: [0.00017652779446273846, 3.550772854555271e-5]
bcs_losses: [1.9922290220867127e-5, 2.3546393957877464e-5, 5.4315904061544124e-5, 2.828282724636963e-5, 2.8313449663713296e-5, 5.991050932412083e-5]
loss: 0.0003515745950111016
pde_losses: [0.00015992677754380317, 4.5759253495556035e-5]
bcs_losses: [1.6742504538460517e-5, 1.310040770856951e-5, 3.894851652068872e-5, 2.1103174255202947e-5, 1.6302470813980393e-5, 3.9691490134840256e-5]
loss: 0.00030924659597715717
pde_losses: [0.0001465994606067391, 4.962236108310983e-5]
bcs_losses: [1.3494090083946795e-5, 1.1166027333804214e-5, 2.5688011682754284e-5, 2.804205326915828e-5, 1.1904179591501035e-5, 2.273041232614361e-5]
loss: 0.00028887138070732743
pde_losses: [0.0001331418664771183, 4.5633221477721676e-5]
bcs_losses: [1.4192404345721699e-5, 1.5026184710790232e-5, 2.6875167470260857e-5, 2.0370003401950745e-5, 1.1469626103560632e-5, 2.2162906720203262e-5]
loss: 0.0002693698477028307
pde_losses: [0.00013079721349970898, 3.4443316121802524e-5]
bcs_losses: [1.6597729724058727e-5, 9.180939616037527e-6, 2.54884232738727e-5, 1.832657131483781e-5, 1.0191412401047927e-5, 2.4344241751464536e-5]
loss: 0.0002622025539208361
pde_losses: [0.00013121557531536456, 2.9670703262249432e-5]
bcs_losses: [1.5662404963130026e-5, 7.695786924709878e-6, 2.1511323569507567e-5, 2.4319732223569848e-5, 1.0038556403229918e-5, 2.208847125907482e-5]
loss: 0.0002528771907799438
pde_losses: [0.00012901292895405538, 2.7245159285742386e-5]
bcs_losses: [1.6731194461225188e-5, 6.723666100702203e-6, 2.2932748098746972e-5, 1.4644930929428834e-5, 9.420163547025724e-6, 2.6166399403017132e-5]
loss: 0.00023880856619839407
pde_losses: [0.0001199517559233119, 2.4732802015575432e-5]
bcs_losses: [1.7125590793172646e-5, 6.137148810259649e-6, 1.8695206661773675e-5, 1.863415423150935e-5, 9.027598895296989e-6, 2.4504308867494438e-5]
loss: 0.00021751942021677831
pde_losses: [9.73130552662419e-5, 3.047306841441449e-5]
bcs_losses: [1.6444322001477674e-5, 6.017852609242184e-6, 1.4739290813161057e-5, 2.1973978616059724e-5, 8.979076366810013e-6, 2.1578776129371294e-5]
loss: 0.00020126478311075605
pde_losses: [8.056359610164266e-5, 3.079112505027613e-5]
bcs_losses: [1.85944069325284e-5, 5.954471464099625e-6, 1.3317995405367054e-5, 2.0439616004498016e-5, 8.054254260920723e-6, 2.3549317891423445e-5]
loss: 0.00017324640942693157
pde_losses: [4.053831020333098e-5, 3.5841038321849686e-5]
bcs_losses: [2.5383186728933952e-5, 6.323361107346805e-6, 1.2965845683687661e-5, 1.2493913634890279e-5, 7.968101625582443e-6, 3.173265212130976e-5]
loss: 0.00016528661472876426
pde_losses: [2.2351287193430218e-5, 4.0157803581011197e-5]
bcs_losses: [2.896927947204557e-5, 1.1035003254598292e-5, 7.255068643421172e-6, 2.1564847149256156e-5, 8.275061214323613e-6, 2.5678264220678062e-5]
loss: 0.00014798429932222313
pde_losses: [2.476101683515245e-5, 3.215770203608232e-5]
bcs_losses: [2.8304186436903447e-5, 4.2990252275021e-6, 9.007175995987373e-6, 1.740232815334503e-5, 8.487792969115853e-6, 2.3565071668134557e-5]
loss: 0.00013880858587628691
pde_losses: [2.398016294187674e-5, 3.18985865188161e-5]
bcs_losses: [2.6663367706565797e-5, 1.295718493591186e-6, 9.425668196610306e-6, 1.7086141446112264e-5, 8.752070109625608e-6, 1.9706870463088925e-5]
loss: 0.00013476409638601402
pde_losses: [2.4707805417505157e-5, 3.146888142646527e-5]
bcs_losses: [2.3592879479667163e-5, 8.873489956629485e-7, 1.0485165328162056e-5, 1.6662137355636763e-5, 9.009140294887349e-6, 1.7950738088027338e-5]
loss: 0.00013222385531156766
pde_losses: [2.4381242517085216e-5, 3.10520990798374e-5]
bcs_losses: [2.1775801763273852e-5, 8.60017529901851e-7, 1.1350207643257631e-5, 1.5086712482335116e-5, 9.04297719644644e-6, 1.867479709943016e-5]
loss: 0.0001259098260702157
pde_losses: [2.3248782675735893e-5, 2.8248820079064726e-5]
bcs_losses: [1.7920835441657035e-5, 8.262967082789226e-7, 1.3409278594143658e-5, 1.1939151007532239e-5, 9.073610456662495e-6, 2.124305110714076e-5]
loss: 0.00011998154080083958
pde_losses: [2.2148296343543274e-5, 2.3981935047686604e-5]
bcs_losses: [1.601163102026092e-5, 9.70040302567758e-7, 1.4708136557614916e-5, 9.929373833355188e-6, 8.98254598779668e-6, 2.3249581708014254e-5]
loss: 0.00011184745281544236
pde_losses: [1.9808431184184827e-5, 1.8592018225364995e-5]
bcs_losses: [1.48952061700563e-5, 1.2717075171470446e-6, 1.515048972397663e-5, 9.722692242278497e-6, 8.65616011419596e-6, 2.3750747638238094e-5]
loss: 0.00010689285529421243
pde_losses: [1.751477983069521e-5, 1.6908167624609005e-5]
bcs_losses: [1.5362277539165098e-5, 1.5839665770983585e-6, 1.4172486868688944e-5, 1.1269953690110816e-5, 8.276337918513175e-6, 2.1804885245331835e-5]
loss: 0.00010413521050085147
pde_losses: [1.541395905670141e-5, 1.6353106074440532e-5]
bcs_losses: [1.696685769454749e-5, 1.8467193350286337e-6, 1.2625088750045822e-5, 1.3951397949009502e-5, 8.077182477408808e-6, 1.890089916366928e-5]
loss: 0.00010255489075039264
pde_losses: [1.381878749727955e-5, 1.5808736631151495e-5]
bcs_losses: [1.8650272824156722e-5, 1.9620878297497216e-6, 1.118401359214945e-5, 1.635438425632503e-5, 8.012870482075841e-6, 1.676373763750483e-5]
loss: 0.00010088605845713122
pde_losses: [1.2509267534353586e-5, 1.516673271070978e-5]
bcs_losses: [1.9573874160627765e-5, 2.070506889891979e-6, 9.998987552494048e-6, 1.808640719462225e-5, 8.045941336635785e-6, 1.5434341077796027e-5]
loss: 9.782043300116611e-5
pde_losses: [1.0637976445035757e-5, 1.4366702098610215e-5]
bcs_losses: [1.996967213101784e-5, 2.3507816812858833e-6, 8.470163534649784e-6, 1.9794032228931625e-5, 8.133788382578294e-6, 1.4097316499056703e-5]
loss: 9.255538368385332e-5
pde_losses: [8.92072056457057e-6, 1.3149891375485114e-5]
bcs_losses: [1.88791003743269e-5, 2.81893598428738e-6, 7.044264145167906e-6, 2.0212235211139058e-5, 8.270455656915474e-6, 1.3259780371960926e-5]
loss: 8.380134684317426e-5
pde_losses: [9.191561172594009e-6, 1.044983997219916e-5]
bcs_losses: [1.5943663858904544e-5, 3.2583721510075897e-6, 6.101448946853425e-6, 1.7251452022025107e-5, 8.362713112900872e-6, 1.3242295606689554e-5]
loss: 7.662506809228072e-5
pde_losses: [1.1907943727830295e-5, 7.885577984308192e-6]
bcs_losses: [1.3724204000310865e-5, 2.7692871629791135e-6, 5.88027644204395e-6, 1.2700111642102942e-5, 8.245906157667013e-6, 1.3511760975038354e-5]
loss: 7.289187959150868e-5
pde_losses: [1.4172211190184687e-5, 7.210043110409242e-6]
bcs_losses: [1.3409788152545462e-5, 1.7943265778670046e-6, 6.225222504137552e-6, 9.072446659845004e-6, 7.744384268804167e-6, 1.326345712771556e-5]
loss: 7.122982672440603e-5
pde_losses: [1.4543450838074538e-5, 7.388210148137265e-6]
bcs_losses: [1.3938701653472089e-5, 1.0797105630438512e-6, 6.738353908460839e-6, 7.303867841407334e-6, 7.498854584524234e-6, 1.2738677187285889e-5]
loss: 7.093884955423748e-5
pde_losses: [1.4069844329452366e-5, 7.923909929950192e-6]
bcs_losses: [1.4330239240644426e-5, 9.391020701674382e-7, 7.144131521790634e-6, 6.925491859223187e-6, 7.42179978998552e-6, 1.2184330813023715e-5]
loss: 7.084573110882615e-5
pde_losses: [1.3113459974491606e-5, 8.23010315193015e-6]
bcs_losses: [1.4867009184386162e-5, 9.197760956858665e-7, 7.879835258629563e-6, 6.055877537946874e-6, 7.409585166976269e-6, 1.2370084738779661e-5]
loss: 7.039652487277959e-5
pde_losses: [1.272978177881254e-5, 7.939595563069274e-6]
bcs_losses: [1.4752534768121452e-5, 9.581107134482333e-7, 7.526387403042114e-6, 6.7993594489929555e-6, 7.4781315790821655e-6, 1.2212623618210863e-5]
loss: 6.99453101365139e-5
pde_losses: [1.2454186352359347e-5, 7.857021708104752e-6]
bcs_losses: [1.4654632867823806e-5, 9.692639818064722e-7, 7.3423306007792144e-6, 7.05537890228042e-6, 7.534116653604749e-6, 1.207837906975514e-5]
loss: 6.813450351288848e-5
pde_losses: [1.0739213818440779e-5, 8.106500814399622e-6]
bcs_losses: [1.4735511133212866e-5, 9.373907090072051e-7, 6.982733543948012e-6, 7.3855351184683774e-6, 7.741856099526111e-6, 1.1505762275885514e-5]
loss: 6.55437733246486e-5
pde_losses: [8.586466869172212e-6, 9.161411998523128e-6]
bcs_losses: [1.5239363616826767e-5, 7.65136934742542e-7, 6.61236575544623e-6, 6.624671997455774e-6, 7.927121664185803e-6, 1.0627234488296146e-5]
loss: 6.227463091809226e-5
pde_losses: [6.421689167071443e-6, 1.1254482526660805e-5]
bcs_losses: [1.656444801034382e-5, 4.78289283609874e-7, 6.09603448248075e-6, 4.450213411144575e-6, 8.011893479005817e-6, 8.997580557775187e-6]
loss: 6.053718634633239e-5
pde_losses: [5.1535340791801965e-6, 1.2228930354431277e-5]
bcs_losses: [1.8065261019474258e-5, 4.973948445921807e-7, 5.998655510935267e-6, 2.769488910440562e-6, 7.958879125423484e-6, 7.865042501855163e-6]
loss: 5.962499739879254e-5
pde_losses: [4.919712732815747e-6, 1.1383356967438972e-5]
bcs_losses: [1.8731975316805584e-5, 5.820333641967708e-7, 6.10477180788649e-6, 2.2997903735799276e-6, 7.922320324742171e-6, 7.681036511326875e-6]
loss: 5.8023041659191304e-5
pde_losses: [5.164019035259288e-6, 9.864301316121067e-6]
bcs_losses: [1.954417937519624e-5, 6.653323433888043e-7, 5.807865034186577e-6, 1.737213268237971e-6, 7.988816857207569e-6, 7.2513144295937964e-6]
loss: 5.6350393074261536e-5
pde_losses: [5.295672708056303e-6, 8.203445483969088e-6]
bcs_losses: [1.9578828573029684e-5, 7.562680547775441e-7, 5.755034932278469e-6, 1.7216287717666368e-6, 8.257525425782638e-6, 6.781989124601175e-6]
loss: 5.368306633668914e-5
pde_losses: [4.855565470226517e-6, 7.106904690850505e-6]
bcs_losses: [1.9007785349243295e-5, 7.651490428376111e-7, 5.362342998532645e-6, 1.7831768884337274e-6, 8.993735326344279e-6, 5.808406570220557e-6]
loss: 5.164781174193743e-5
pde_losses: [4.068742372298369e-6, 6.271756979136575e-6]
bcs_losses: [1.7106705455039547e-5, 5.457684688423627e-7, 5.565739535380072e-6, 2.5378394038682176e-6, 9.923831287219483e-6, 5.627428240152802e-6]
loss: 5.094773125703633e-5
pde_losses: [3.3808361047061858e-6, 6.167302388280866e-6]
bcs_losses: [1.6638080654474333e-5, 4.703528041398625e-7, 5.7706422441846835e-6, 2.6990537415682506e-6, 1.0337684696717873e-5, 5.483778622964268e-6]
loss: 5.047325772071771e-5
pde_losses: [3.3269134898984885e-6, 5.693891610296999e-6]
bcs_losses: [1.5976809182318354e-5, 4.3161429774558287e-7, 6.05791835509949e-6, 2.728673020443167e-6, 1.0152067346199183e-5, 6.105370418716448e-6]
loss: 5.032447276064167e-5
pde_losses: [3.2074176878836183e-6, 5.676324768590487e-6]
bcs_losses: [1.609937160310022e-5, 4.661771789122159e-7, 6.144072059912037e-6, 2.602461031389802e-6, 1.0005566059370684e-5, 6.12308237148261e-6]
loss: 5.0132155524764745e-5
pde_losses: [3.029828339842596e-6, 5.661195986746863e-6]
bcs_losses: [1.628508654672204e-5, 5.435164828820705e-7, 6.229884197807748e-6, 2.442690898673846e-6, 9.995302154135103e-6, 5.944650917954485e-6]
loss: 4.9986869637008955e-5
pde_losses: [2.9537504947276388e-6, 5.637650343783912e-6]
bcs_losses: [1.6382639257684185e-5, 5.928781565740849e-7, 6.185832486095579e-6, 2.4466653532477917e-6, 1.0105278874192604e-5, 5.682174670703167e-6]
loss: 4.980322449326526e-5
pde_losses: [2.9244693275608463e-6, 5.597225805193829e-6]
bcs_losses: [1.6447704969954747e-5, 6.469791410088048e-7, 6.022066007006329e-6, 2.537113946760462e-6, 1.0340528202778665e-5, 5.2871370930015775e-6]
loss: 4.962173237137121e-5
pde_losses: [2.9481004069837387e-6, 5.572637792013486e-6]
bcs_losses: [1.64822467442547e-5, 6.709024378351215e-7, 5.791194404461216e-6, 2.7090872709986347e-6, 1.050601568482022e-5, 4.9415476300040855e-6]
loss: 4.9343601390460514e-5
pde_losses: [3.039771623058776e-6, 5.550016323102162e-6]
bcs_losses: [1.6513565932035466e-5, 7.050129577708647e-7, 5.436268801681551e-6, 2.9332575040689438e-6, 1.063192039432891e-5, 4.53378785441384e-6]
loss: 4.888057349275652e-5
pde_losses: [3.238018880288005e-6, 5.514987667512479e-6]
bcs_losses: [1.654667465013281e-5, 7.617257970566364e-7, 4.9352405674162016e-6, 3.232434509792519e-6, 1.0603843468186314e-5, 4.047647952371547e-6]
loss: 4.7829119982752675e-5
pde_losses: [3.6016395532983015e-6, 5.334402157338501e-6]
bcs_losses: [1.6447473241221747e-5, 8.965537461612501e-7, 4.212896897502465e-6, 3.576596254216247e-6, 1.0304208863615585e-5, 3.4553492693985896e-6]
loss: 4.613043155836292e-5
pde_losses: [4.175274359755059e-6, 5.3826524973828285e-6]
bcs_losses: [1.5859472033454023e-5, 1.1540961424796162e-6, 3.2637477973010434e-6, 4.031964530428375e-6, 9.441125784471331e-6, 2.8220984130906377e-6]
loss: 4.316885001179601e-5
pde_losses: [4.561506033988992e-6, 6.356905669733345e-6]
bcs_losses: [1.3780335133636444e-5, 1.4339143924195725e-6, 2.477863891995715e-6, 4.5135296468988394e-6, 7.602250916389021e-6, 2.44254432673408e-6]
loss: 4.2510280564102785e-5
pde_losses: [4.654913136562897e-6, 6.128987179069427e-6]
bcs_losses: [1.3373445996967038e-5, 1.533326222090143e-6, 2.4473041429521123e-6, 4.686155981703903e-6, 7.2415777577973815e-6, 2.444570146959885e-6]
loss: 4.081522942826005e-5
pde_losses: [3.6237891181802393e-6, 8.831595649867145e-6]
bcs_losses: [8.252651113098396e-6, 1.5719915561877815e-6, 4.923732240934473e-6, 6.202932469798679e-6, 4.398036270014198e-6, 3.0105010101791376e-6]
loss: 3.562756257400668e-5
pde_losses: [2.8433511410547965e-6, 5.587286242636155e-6]
bcs_losses: [7.905437742550849e-6, 1.0889501042977246e-6, 4.152264650398321e-6, 5.48658864518075e-6, 5.573787293217119e-6, 2.989896754670965e-6]
loss: 3.334137267765248e-5
pde_losses: [2.532195078191596e-6, 4.785715537396078e-6]
bcs_losses: [6.54388210846404e-6, 9.922789695588788e-7, 4.12090228898966e-6, 5.645359537965287e-6, 5.8222523907232344e-6, 2.898786766363708e-6]
loss: 3.280174963281243e-5
pde_losses: [2.2838710133004846e-6, 4.8552401123037345e-6]
bcs_losses: [4.445533194359513e-6, 6.581163827680475e-7, 6.625133946174974e-6, 5.30016681353498e-6, 5.960057932116155e-6, 2.6736302382545375e-6]
loss: 3.0963097327282005e-5
pde_losses: [2.004024427261226e-6, 4.0898955616331214e-6]
bcs_losses: [5.2000518196804404e-6, 6.766578754311232e-7, 4.971111159251843e-6, 5.2752320852908315e-6, 6.1463017718699025e-6, 2.59982262686352e-6]
loss: 3.069967170192012e-5
pde_losses: [1.9353612905186896e-6, 4.164416358356829e-6]
bcs_losses: [4.919775300942041e-6, 6.6382402465195e-7, 5.3018121025796076e-6, 5.021669896484394e-6, 6.21720710111248e-6, 2.4756056272741247e-6]
loss: 3.0469823247461008e-5
pde_losses: [1.8423111664299327e-6, 4.274828291597597e-6]
bcs_losses: [4.481586507225237e-6, 6.134978620594897e-7, 5.835652965673234e-6, 4.7900906999288795e-6, 6.27650718465884e-6, 2.3553485698877986e-6]
loss: 3.0423095174457863e-5
pde_losses: [1.8459661445660605e-6, 4.487825037554701e-6]
bcs_losses: [4.337599816934916e-6, 6.208870395788385e-7, 5.979605233856798e-6, 4.614735652643347e-6, 6.243849793550768e-6, 2.2926264557724346e-6]
loss: 3.0384323876319314e-5
pde_losses: [1.858286247658103e-6, 4.481111810438931e-6]
bcs_losses: [4.394454394750967e-6, 6.285176485529126e-7, 5.892978640812061e-6, 4.61134167042175e-6, 6.2104586365953765e-6, 2.3071748270892137e-6]
loss: 3.033055602784924e-5
pde_losses: [1.8766992844433003e-6, 4.532352931863639e-6]
bcs_losses: [4.4724417785095525e-6, 6.454866796350574e-7, 5.7168617929039025e-6, 4.595875333441658e-6, 6.163658687960165e-6, 2.327179539091964e-6]
loss: 3.027821328677745e-5
pde_losses: [1.8723607969285647e-6, 4.560657852996977e-6]
bcs_losses: [4.513493254794972e-6, 6.494293427284848e-7, 5.58185100075003e-6, 4.5707928286449116e-6, 6.184576306326299e-6, 2.3450519036072095e-6]
loss: 3.0193165757725536e-5
pde_losses: [1.8458376176792221e-6, 4.552223653925116e-6]
bcs_losses: [4.551530111699333e-6, 6.297362893622976e-7, 5.4206504776209085e-6, 4.55317637710371e-6, 6.27526146053261e-6, 2.364749769802334e-6]
loss: 3.0114565898342445e-5
pde_losses: [1.8086286241782865e-6, 4.478830322228604e-6]
bcs_losses: [4.557359265679768e-6, 5.878533339861634e-7, 5.366393040447174e-6, 4.545442480629222e-6, 6.405289243078346e-6, 2.364769588114881e-6]
loss: 3.004134440057496e-5
pde_losses: [1.761276125919128e-6, 4.329356137093944e-6]
bcs_losses: [4.564103050530138e-6, 5.441792770978672e-7, 5.409136778134067e-6, 4.55349278043262e-6, 6.526945051933737e-6, 2.35285519943346e-6]
loss: 2.996825722057019e-5
pde_losses: [1.7087125677095673e-6, 4.167964341427349e-6]
bcs_losses: [4.587317704439003e-6, 5.21308743156893e-7, 5.489388065293282e-6, 4.548015780243549e-6, 6.60971394967086e-6, 2.335836068629686e-6]
loss: 2.984995349241353e-5
pde_losses: [1.6401822226670423e-6, 3.952833934604559e-6]
bcs_losses: [4.637205817411444e-6, 5.151849771221605e-7, 5.594875040144782e-6, 4.518754161192091e-6, 6.6737414699768e-6, 2.317175869294652e-6]
loss: 2.9599352632021584e-5
pde_losses: [1.5449457518710458e-6, 3.6555758175398133e-6]
bcs_losses: [4.704129626082613e-6, 5.375769077442022e-7, 5.715621746721519e-6, 4.4122998863698916e-6, 6.7411894110147525e-6, 2.2880134846777464e-6]
loss: 2.9014197032956513e-5
pde_losses: [1.4162868750672532e-6, 3.184589608472071e-6]
bcs_losses: [4.729064730189864e-6, 6.201267070727619e-7, 5.840162858450772e-6, 4.16695339158447e-6, 6.821272563397002e-6, 2.2357402987223187e-6]
loss: 2.814184107098689e-5
pde_losses: [1.3197851706229216e-6, 2.9031985401862767e-6]
bcs_losses: [4.448264933101016e-6, 8.449739365355631e-7, 5.819447109942309e-6, 3.6971307068325812e-6, 7.0023546523651545e-6, 2.1066860214010672e-6]
loss: 2.7132393171794215e-5
pde_losses: [1.3903997101628566e-6, 2.479934842561615e-6]
bcs_losses: [4.126564004633647e-6, 9.310505514136488e-7, 5.381051190196489e-6, 3.6067717407417937e-6, 7.177330234195649e-6, 2.0392908978885194e-6]
loss: 2.6634322661737565e-5
pde_losses: [1.41394455349161e-6, 2.816073456554456e-6]
bcs_losses: [3.940829199256766e-6, 1.1657114391735594e-6, 4.677498456572243e-6, 3.287443001641856e-6, 7.3016899014427725e-6, 2.0311326536043018e-6]
loss: 2.6333814189646686e-5
pde_losses: [1.3717296152905506e-6, 2.6213770086530887e-6]
bcs_losses: [4.054427431271083e-6, 1.1336431579190068e-6, 4.69549203291168e-6, 3.278387383558457e-6, 7.102418378277534e-6, 2.076339181765283e-6]
loss: 2.61697261934946e-5
pde_losses: [1.341584841514363e-6, 2.5232017354296476e-6]
bcs_losses: [3.897301755042935e-6, 1.1816433332332606e-6, 4.64220310332698e-6, 3.350266452211288e-6, 7.145217162107604e-6, 2.088307810628526e-6]
loss: 2.5956439734509375e-5
pde_losses: [1.3213870215237594e-6, 2.4852539199265532e-6]
bcs_losses: [3.5049468062245856e-6, 1.3554999247995929e-6, 4.366165650478325e-6, 3.488524220377006e-6, 7.30820489797718e-6, 2.126457293202377e-6]
loss: 2.5791827746437756e-5
pde_losses: [1.3023631397621599e-6, 2.3016767295848344e-6]
bcs_losses: [3.5508017349964594e-6, 1.3051670641205675e-6, 4.420164569178165e-6, 3.5469231095072967e-6, 7.24204857510865e-6, 2.1226828241796214e-6]
loss: 2.561540143181637e-5
pde_losses: [1.3321777553481422e-6, 2.1763070292154824e-6]
bcs_losses: [3.5621146505786765e-6, 1.327325129800933e-6, 4.417860747596644e-6, 3.4742754863586085e-6, 7.218324219718126e-6, 2.1070164131997562e-6]
loss: 2.54342700515759e-5
pde_losses: [1.4546577413011021e-6, 2.0657509191087698e-6]
bcs_losses: [3.4654351687728015e-6, 1.4797445226808215e-6, 4.20646930437931e-6, 3.3274289612192554e-6, 7.34219460156938e-6, 2.0925888325444633e-6]
loss: 2.534030320760978e-5
pde_losses: [1.468621580197387e-6, 2.0247469739899277e-6]
bcs_losses: [3.5022435643945854e-6, 1.495503718009171e-6, 4.216242177253216e-6, 3.217529784443132e-6, 7.351564603517415e-6, 2.0638508058049447e-6]
loss: 2.5281519930423165e-5
pde_losses: [1.5039666826039196e-6, 1.8965826232522832e-6]
bcs_losses: [3.4919859987956015e-6, 1.535001111472018e-6, 4.244368218751805e-6, 3.186322907889298e-6, 7.357177457881366e-6, 2.066114929776875e-6]
loss: 2.5261686625883515e-5
pde_losses: [1.6139574959074948e-6, 1.873978341847034e-6]
bcs_losses: [3.3148458938037554e-6, 1.6559751760801662e-6, 4.089144944920511e-6, 3.1649893911104255e-6, 7.466715985853111e-6, 2.0820793963610145e-6]
loss: 2.5257993978179802e-5
pde_losses: [1.6115493253166872e-6, 1.8712259792041378e-6]
bcs_losses: [3.317456539167476e-6, 1.654733677758102e-6, 4.090068029031144e-6, 3.166749445503519e-6, 7.463768386985684e-6, 2.082442595213052e-6]
loss: 2.525560941719683e-5
pde_losses: [1.6115560810861168e-6, 1.8687943926780757e-6]
bcs_losses: [3.3184542687758265e-6, 1.6547886443511006e-6, 4.089542659270714e-6, 3.1674945717526193e-6, 7.4622095753138215e-6, 2.0827692239685534e-6]
loss: 2.525235302223463e-5
pde_losses: [1.6113542972126032e-6, 1.8668774709014911e-6]
bcs_losses: [3.3193378282095286e-6, 1.655052724819522e-6, 4.087644019227348e-6, 3.1688124470174794e-6, 7.459894343751831e-6, 2.0833798910948205e-6]
loss: 2.5246333928870116e-5
pde_losses: [1.6108122653715754e-6, 1.8634334776500614e-6]
bcs_losses: [3.3199451314546264e-6, 1.655581827712282e-6, 4.0844375361932624e-6, 3.1708795074832186e-6, 7.456906139403825e-6, 2.0843380436012644e-6]
loss: 2.5241498259678018e-5
pde_losses: [1.6118440164106005e-6, 1.8606097096869886e-6]
bcs_losses: [3.3196049488917234e-6, 1.6557098612864947e-6, 4.081756135508206e-6, 3.172772468683459e-6, 7.45394956718696e-6, 2.085251552023584e-6]
loss: 2.5240749643439645e-5
pde_losses: [1.6121145441375237e-6, 1.8599473812164724e-6]
bcs_losses: [3.319621514656254e-6, 1.6557741823754878e-6, 4.0812720295659055e-6, 3.1730558796611235e-6, 7.453576832136754e-6, 2.085387279690123e-6]
loss: 2.5239352648461417e-5
pde_losses: [1.611165931156518e-6, 1.8600964798963767e-6]
bcs_losses: [3.3193815848460055e-6, 1.6558275656304035e-6, 4.0808003736220995e-6, 3.173365972087205e-6, 7.453169654904254e-6, 2.085545086318556e-6]
loss: 2.523824684320389e-5
pde_losses: [1.6120091829915106e-6, 1.8589123797076615e-6]
bcs_losses: [3.319417591116802e-6, 1.6559489093164131e-6, 4.079842530084672e-6, 3.1739121461831446e-6, 7.4523982627784545e-6, 2.085805841025235e-6]
loss: 2.5237933667186702e-5
pde_losses: [1.6118354562341909e-6, 1.8587986013933526e-6]
bcs_losses: [3.3194054557986745e-6, 1.6559481371143945e-6, 4.079831309657562e-6, 3.1739200135989563e-6, 7.452384614949555e-6, 2.085810078440015e-6]
loss: 2.5237805817248993e-5
pde_losses: [1.61168035626975e-6, 1.8588258514346236e-6]
bcs_losses: [3.319405455835122e-6, 1.655948137110673e-6, 4.079831309592774e-6, 3.173920013655779e-6, 7.452384614884215e-6, 2.0858100784660574e-6]
loss: 2.5237805817248993e-5
pde_losses: [1.61168035626975e-6, 1.8588258514346236e-6]
bcs_losses: [3.319405455835122e-6, 1.655948137110673e-6, 4.079831309592774e-6, 3.173920013655779e-6, 7.452384614884215e-6, 2.0858100784660574e-6]
loss: 2.5237805817248993e-5
pde_losses: [1.61168035626975e-6, 1.8588258514346236e-6]
bcs_losses: [3.319405455835122e-6, 1.655948137110673e-6, 4.079831309592774e-6, 3.173920013655779e-6, 7.452384614884215e-6, 2.0858100784660574e-6]

ps[1]

ps[2]