Linear parabolic system of PDEs

We can use NeuralPDE to solve the linear parabolic system of PDEs:

\[\begin{aligned} \frac{\partial u}{\partial t} &= a * \frac{\partial^2 u}{\partial x^2} + b_1 u + c_1 w \\ \frac{\partial w}{\partial t} &= a * \frac{\partial^2 w}{\partial x^2} + b_2 u + c_2 w \\ \end{aligned}\]

with initial and boundary conditions:

\[\begin{aligned} u(0, x) = \frac{b_1 - \lambda_2}{b_2 (\lambda_1 - \lambda_2)} \cdot cos(\frac{x}{a}) - \frac{b_1 - \lambda_1}{b_2 (\lambda_1 - \lambda_2)} \cdot cos(\frac{x}{a}) \\ w(0, x) = 0 \\ u(t, 0) = \frac{b_1 - \lambda_2}{b_2 (\lambda_1 - \lambda_2)} \cdot e^{\lambda_1t} - \frac{b_1 - \lambda_1}{b_2 (\lambda_1 - \lambda_2)} \cdot e^{\lambda_2t} \\ w(t, 0) = \frac{e^{\lambda_1}-e^{\lambda_2}}{\lambda_1 - \lambda_2} \\ u(t, 1) = \frac{b_1 - \lambda_2}{b_2 (\lambda_1 - \lambda_2)} \cdot e^{\lambda_1t} \cdot cos(\frac{x}{a}) - \frac{b_1 - \lambda_1}{b_2 (\lambda_1 - \lambda_2)} \cdot e^{\lambda_2t} * cos(\frac{x}{a}) \\ w(t, 1) = \frac{e^{\lambda_1} cos(\frac{x}{a})-e^{\lambda_2}cos(\frac{x}{a})}{\lambda_1 - \lambda_2} \end{aligned}\]

with a physics-informed neural network.

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

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

# Constants
a  = 1
b1 = 4
b2 = 2
c1 = 3
c2 = 1
λ1 = (b1 + c2 + sqrt((b1 + c2)^2 + 4 * (b1 * c2 - b2 * c1))) / 2
λ2 = (b1 + c2 - sqrt((b1 + c2)^2 + 4 * (b1 * c2 - b2 * c1))) / 2

# Analytic solution
θ(t, x) = exp(-t) * cos(x / a)
u_analytic(t, x) = (b1 - λ2) / (b2 * (λ1 - λ2)) * exp(λ1 * t) * θ(t, x) - (b1 - λ1) / (b2 * (λ1 - λ2)) * exp(λ2 * t) * θ(t, x)
w_analytic(t, x) = 1 / (λ1 - λ2) * (exp(λ1 * t) * θ(t, x) - exp(λ2 * t) * θ(t, x))

# Second-order constant-coefficient linear parabolic system
eqs = [Dt(u(x, t)) ~ a * Dxx(u(x, t)) + b1 * u(x, t) + c1 * w(x, t),
       Dt(w(x, t)) ~ a * Dxx(w(x, t)) + b2 * u(x, t) + c2 * w(x, t)]

# 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 = [x ∈ Interval(0.0, 1.0),
           t ∈ 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=5000)

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$i, analytic");
    p2 = plot(ts, xs, u_predict[i], linetype=:contourf, title="predict");
    p3 = plot(ts, xs, diff_u[i], linetype=:contourf, title="error");
    plot(p1, p2, p3)
    savefig("sol_u$i")
end

loss: 1327.877458345974
pde_losses: [5.560486564525849, 0.9468986275202422]
bcs_losses: [0.8394863728713055, 15.74476867890414, 681.5609202809127, 321.1332461558723, 175.2379082932955, 126.85374337207199]
loss: 1266.739383335777
pde_losses: [29.42475521743994, 18.186446405908423]
bcs_losses: [3.566855339633104, 8.506969725676441, 647.2103551577914, 292.38426915712375, 159.6572660743807, 107.80246625782313]
loss: 1253.1806525285294
pde_losses: [31.603849302524768, 5.6154372413160445]
bcs_losses: [0.9945455744441472, 0.5788323890524375, 741.1836234263608, 209.2450178110803, 205.31503153590904, 58.64431524784185]
loss: 1204.1607377112782
pde_losses: [29.29665798462503, 6.9191246084804225]
bcs_losses: [2.5921446546924045, 1.3771870899595435, 670.5670675222818, 230.1949179991576, 187.21096333464928, 76.00267451743196]
loss: 1189.2488386443945
pde_losses: [41.68329447840366, 11.762342679740938]
bcs_losses: [1.3208808165443013, 1.8366294556811287, 646.9282889880403, 232.53477828198305, 175.24337158388093, 77.93925236012025]
loss: 1180.9226153092873
pde_losses: [75.95534801556131, 22.802191374667956]
bcs_losses: [6.754358065192185, 0.2678624119326197, 615.6824190670009, 210.462767287136, 184.14976127514734, 64.84790781264871]
loss: 1168.310564794837
pde_losses: [75.67307567901757, 24.322390343941795]
bcs_losses: [6.143173777322718, 0.06590144270995485, 611.924020647712, 202.9330116094594, 186.95910080152336, 60.28989049315018]
loss: 1164.896806838237
pde_losses: [76.99338262562085, 25.213035895889327]
bcs_losses: [5.689935005577988, 0.05936358424429729, 608.8369440436071, 202.29409701153196, 185.98050222973075, 59.82954644203468]
loss: 1160.7085939570186
pde_losses: [36.066961180864304, 13.542721730669115]
bcs_losses: [4.3658447242251, 0.0379601975287178, 638.2869477769237, 205.09162746309573, 203.29071076235525, 60.02582012135677]
loss: 1138.4347683875849
pde_losses: [43.06182865641484, 18.402829537829252]
bcs_losses: [3.318380284767671, 0.04234996679950344, 618.6728495043936, 205.28530575121428, 191.13125989260155, 58.51996479356429]
loss: 1105.7534830224029
pde_losses: [55.48108220878623, 35.67078465874664]
bcs_losses: [4.957794340883202, 0.006978139638095135, 561.3037427798938, 214.4846643309064, 172.26103709363483, 61.587399469913564]
loss: 1103.8968382418373
pde_losses: [53.33733336972636, 36.01109173333425]
bcs_losses: [5.306837038690313, 0.007783803631374324, 559.69879946721, 214.95856511236718, 172.8111521792052, 61.76527553767249]
loss: 1083.1680784017217
pde_losses: [53.73110377224689, 51.04310915695637]
bcs_losses: [6.781874663304832, 0.06609699529659736, 512.6911374696832, 221.58497208568647, 172.7164502862371, 64.55333397231038]
loss: 1070.384918040834
pde_losses: [47.2055828249417, 59.61533151465339]
bcs_losses: [7.485602727667242, 0.16516742241548432, 492.9183954175279, 225.6594038235731, 170.92669676058472, 66.40873754947049]
loss: 1035.8801822366102
pde_losses: [44.22853080094099, 97.98833812395877]
bcs_losses: [6.010504720944043, 0.49985631497820165, 402.9271605122993, 233.75574161065805, 180.36985153426963, 70.10019861856131]
loss: 1025.4682447704831
pde_losses: [42.89911118381067, 117.72209987980735]
bcs_losses: [4.764593979888184, 0.6261569199433553, 371.97396785400497, 235.5325699463109, 181.0335441462894, 70.9162008604285]
loss: 1020.6794526413173
pde_losses: [26.954122334084442, 115.73819995783552]
bcs_losses: [4.201970419269774, 0.7480815208978263, 361.8274010768972, 235.8199627665977, 204.40587736372055, 70.98383720201421]
loss: 1006.4413205650114
pde_losses: [28.230583462710673, 130.51162025228024]
bcs_losses: [2.3904500817104646, 0.7867635989922335, 342.985379554059, 230.75071218348137, 202.70043485592453, 68.08537657585288]
loss: 986.3539003044668
pde_losses: [36.67255454023689, 150.91694729333148]
bcs_losses: [0.7807453935376185, 0.7844382060927696, 320.88136902530624, 216.27827952483412, 199.77752976495125, 60.262036556176305]
loss: 972.3816991612248
pde_losses: [41.022244405185276, 167.26164768252775]
bcs_losses: [0.32174054075190384, 0.7006034248656607, 310.6518687244579, 201.0108017225213, 198.91788959318583, 52.494903067729155]
loss: 953.9044078459995
pde_losses: [46.50061290299417, 162.27096642898667]
bcs_losses: [0.15776267804029373, 0.8581150590244584, 319.2873813246816, 185.03401193355086, 194.60794852894307, 45.18760898977843]
loss: 927.7636149013913
pde_losses: [29.92053163876076, 155.3529617049781]
bcs_losses: [0.8306296185600643, 0.7728294318938176, 335.9237046779592, 164.95188799272762, 203.22700498705467, 36.78406484945711]
loss: 917.8738120600108
pde_losses: [29.29936016474446, 157.38919306871142]
bcs_losses: [1.2310567551103464, 0.6700514761593749, 338.04929664944655, 156.63931305228817, 200.9416509305415, 33.65388996300899]
loss: 895.248185875402
pde_losses: [30.31067667090038, 157.71229279369288]
bcs_losses: [2.692028477953377, 0.2809862020495714, 338.87593169310725, 142.92943814564168, 192.42645920655892, 30.020372685497904]
loss: 845.1044461881995
pde_losses: [40.28285410098855, 146.80214031669277]
bcs_losses: [12.437345487113559, 0.824177588017335, 342.0155117033364, 99.77670161754803, 175.08484619709066, 27.880869177412247]
loss: 833.9270997088273
pde_losses: [60.153094707384874, 158.64695768873608]
bcs_losses: [16.599692168045458, 3.570924025023941, 352.4616080873393, 78.18631816711826, 142.57574638311382, 21.732758482065577]
loss: 824.7544717976298
pde_losses: [58.200705998426635, 158.1033451290039]
bcs_losses: [17.424808131919402, 4.3502341112958955, 353.8375048666695, 72.9814154397934, 139.6370778559037, 20.219380264617335]
loss: 801.81609328539
pde_losses: [59.48579509526519, 149.7221065127325]
bcs_losses: [18.52471452866074, 6.604721005868818, 357.46972424733013, 61.00317125264406, 131.84354287330265, 17.162317769586064]
loss: 783.3155945758126
pde_losses: [64.55882028455851, 135.36127393252107]
bcs_losses: [19.04362998011738, 8.698201889211779, 359.9068430817005, 51.70239210485107, 129.02282311123048, 15.021610191621852]
loss: 762.9784129016052
pde_losses: [71.17497756998661, 144.03547536291813]
bcs_losses: [18.3538383495767, 11.336669218554064, 344.77159747085886, 44.18000132760072, 115.49736533649079, 13.628488265619286]
loss: 736.0167725812321
pde_losses: [77.08684803311996, 130.67874658679486]
bcs_losses: [17.028042733754358, 14.770140072782066, 341.91966701155434, 35.68890117006837, 107.0580214445121, 11.786405528645995]
loss: 716.8431890495585
pde_losses: [82.0092717592671, 122.64510786298824]
bcs_losses: [16.313026867793447, 17.46590863543695, 334.6916403177406, 30.543392367017447, 102.3013240706679, 10.873517168646895]
loss: 699.4052885525011
pde_losses: [93.63158210553516, 113.64901794728989]
bcs_losses: [14.572548436598348, 20.40437953705989, 323.53043949093336, 26.584738539075204, 96.40882548550597, 10.623757010503354]
loss: 677.9400802735136
pde_losses: [100.90316359026716, 104.47773099388328]
bcs_losses: [15.748855264951255, 23.948733768329188, 296.84512233980524, 22.812141480323902, 103.24805082385119, 9.95628201210237]
loss: 666.126845091952
pde_losses: [101.13099010116085, 102.50814003621642]
bcs_losses: [16.307152942502604, 24.742334206793366, 288.1244817369353, 22.258035560472084, 101.12798568559927, 9.927724822272044]
loss: 643.2148762844243
pde_losses: [100.6994703204911, 97.98434346703054]
bcs_losses: [17.373842322984366, 26.153291415779005, 270.7794251464948, 21.91726367033743, 98.29368460391491, 10.013555337392106]
loss: 616.2104647046194
pde_losses: [99.32662749742327, 95.51781883135203]
bcs_losses: [18.55258358319771, 27.77370892537574, 246.8069811847133, 22.09006570997447, 96.08835821954267, 10.054320753040216]
loss: 606.5548959950611
pde_losses: [96.33239476769107, 96.2003151926464]
bcs_losses: [19.72226926704513, 28.0589755833138, 240.4541740932, 22.181252581365246, 93.64063553895735, 9.96487897084213]
loss: 595.4016052887441
pde_losses: [139.19889974146605, 108.94927822030739]
bcs_losses: [24.300209375941634, 27.520284475995524, 189.43735524236828, 22.258176796932048, 75.27099364803237, 8.466407787700735]
loss: 589.7645188945716
pde_losses: [149.99543022270743, 117.04822312982208]
bcs_losses: [24.98646270193977, 25.925017247092246, 171.35136357625296, 24.29560948523865, 68.99897523523072, 7.163437296287688]
loss: 579.2611685190083
pde_losses: [179.23758896414563, 121.29655224963584]
bcs_losses: [18.75075393197886, 21.558981470645364, 149.00204247087987, 29.92789203318661, 53.51937324113725, 5.967984157398813]
loss: 559.9510344416276
pde_losses: [176.68127759882, 123.53843671900502]
bcs_losses: [17.54693759762344, 18.831620821416983, 135.5876886800477, 32.476773246716675, 49.748656169914824, 5.539643608082969]
loss: 535.0859077929974
pde_losses: [178.22933326204495, 122.4371410871593]
bcs_losses: [15.04389263665394, 15.463301271356654, 120.2623848302375, 34.93625654724678, 43.744235995459015, 4.969362162839383]
loss: 505.59390929736435
pde_losses: [171.32053064296358, 118.04907967448483]
bcs_losses: [11.878733188698844, 11.41911778151379, 109.98174408830846, 39.0528976394348, 38.75755432776981, 5.134251954190222]
loss: 488.7948160284508
pde_losses: [165.51113832043808, 117.4544747503965]
bcs_losses: [10.475698274039674, 10.055516406697436, 102.31272938566602, 40.91999679671315, 37.00051067238962, 5.064751422110335]
loss: 476.5971952170923
pde_losses: [164.95769694000953, 118.32138827599518]
bcs_losses: [8.41821047500143, 8.790630237694202, 95.33275381865661, 42.475331768344496, 33.29681276663484, 5.004370934755993]
loss: 465.58691338834376
pde_losses: [164.94297138896712, 126.73168945720842]
bcs_losses: [5.664443743064776, 6.866753516323185, 78.36597670771043, 42.9999933663215, 34.7041932054315, 5.31089200331682]
loss: 448.883302944956
pde_losses: [174.91030780166795, 110.94001101909107]
bcs_losses: [4.424421247013414, 4.694629901272627, 66.99643920240457, 41.44652798505744, 39.0245549479548, 6.446410840494134]
loss: 432.8468360214516
pde_losses: [161.1161194138885, 112.27124750064944]
bcs_losses: [2.6424709817118694, 3.995228193537041, 58.085563904522296, 44.74937919405953, 44.670481166659286, 5.316345666423692]
loss: 421.17050014112453
pde_losses: [146.72807866356388, 112.20214910253476]
bcs_losses: [1.9687569411725878, 3.477031460766208, 56.07352974089741, 44.506101796549856, 51.70930730492914, 4.505545130710661]
loss: 384.98526547025
pde_losses: [115.50831604057554, 88.33572883505559]
bcs_losses: [0.6981194418411155, 2.246642786689861, 55.9039067489588, 49.13898389136584, 69.88812281314995, 3.265444912613316]
loss: 362.82803782513145
pde_losses: [104.80147816468983, 73.5265089580027]
bcs_losses: [0.5333056649948984, 1.7845810453010957, 54.47893182694738, 46.800909506176275, 77.70965199203256, 3.192670666986685]
loss: 327.8891410574996
pde_losses: [75.8342214569305, 54.7396989710611]
bcs_losses: [0.82150560648097, 1.0451859844057552, 49.98716958549089, 47.771386028020615, 94.7751412800157, 2.91483214509412]
loss: 316.21222937151936
pde_losses: [64.14036122593946, 51.683172301641314]
bcs_losses: [0.9730145663930825, 0.885627308529567, 49.14642051136111, 46.5769906393556, 99.82191721501235, 2.9847256032869085]
loss: 303.9727659749641
pde_losses: [55.541375454921955, 46.252674832640395]
bcs_losses: [1.5271602736748517, 0.7432786597601302, 47.03800961310409, 44.94881111753749, 105.01404598759103, 2.907410035734103]
loss: 291.56325769558646
pde_losses: [55.094314641305445, 33.8958096718503]
bcs_losses: [2.3117574954850824, 0.6560503359824934, 41.754982608388644, 46.456638843921795, 108.95417399721276, 2.4395301014399373]
loss: 274.97944195513793
pde_losses: [44.637754391041966, 24.723061330897576]
bcs_losses: [2.782714316260999, 0.7480461982811016, 44.091932255255685, 47.83122720041047, 108.29211814943129, 1.8725881135588762]
loss: 269.1730293090529
pde_losses: [37.51512723077664, 25.402310771198113]
bcs_losses: [3.177179271715339, 0.8409800429599706, 42.35376953865112, 48.991826742467325, 109.20757512202624, 1.6842605892581208]
loss: 259.64925437813906
pde_losses: [31.36298614116274, 22.45985981368159]
bcs_losses: [3.47299525748268, 0.9951244311599114, 41.734778367055924, 50.44564194007381, 107.7041816550202, 1.4736867725022575]
loss: 248.62631278508914
pde_losses: [20.369412026090615, 22.016658174885222]
bcs_losses: [3.5012438913548083, 1.1595879958547337, 42.41795295564772, 51.28769534712294, 106.6655000475881, 1.208262346545003]
loss: 239.15476039903504
pde_losses: [13.952453346795567, 19.253787008862236]
bcs_losses: [3.6921636143312466, 0.9274603607023533, 44.604545782961374, 51.017570172624424, 104.7932875373874, 0.9134925753704661]
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