# 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]

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: 1970.3022199718623
pde_losses: [317.59457401862693, 73.15704413529066]
bcs_losses: [27.09991415731958, 0.6876622643290335, 919.3882825471293, 244.68078151267684, 312.7753777645987, 74.91858357189126]
loss: 1606.208431468876
pde_losses: [115.66309979252735, 28.84073909849951]
bcs_losses: [16.499976997950554, 0.022987988379102868, 866.8264112031063, 230.1544144636352, 281.3083550191263, 66.89244690565174]
loss: 1545.6437434942627
pde_losses: [104.5560557093244, 18.636744510616154]
bcs_losses: [8.098365782842428, 2.647931992622945, 813.039592416556, 262.8024311321117, 250.4408840362714, 85.42173791391757]
loss: 1408.3136176208702
pde_losses: [41.56080312632251, 1.3084266288514819]
bcs_losses: [0.12043454743113288, 16.223714236109593, 706.1176486374427, 324.8333798933157, 193.49356909704665, 124.65564145435052]
loss: 1373.8024300556244
pde_losses: [23.275908461549207, 0.07066375044866068]
bcs_losses: [0.4397291433525569, 21.14882993837567, 674.5850420229925, 340.5049279921319, 178.6343638263455, 135.14296492042826]
loss: 1343.6565816267514
pde_losses: [8.867570062591115, 1.432649310878812]
bcs_losses: [1.696770688374508, 24.52368727185262, 648.369289006567, 350.26057024283716, 166.70460499440196, 141.8014400492484]
loss: 1304.6715382316593
pde_losses: [1.0635976088955874, 6.656491879403827]
bcs_losses: [3.467857662827351, 24.07670535099525, 624.8256078203169, 347.7730392288638, 156.68625050795652, 140.12198817239994]
loss: 1256.7041148102564
pde_losses: [13.705428978623809, 15.447930765979383]
bcs_losses: [3.528751910873298, 16.216697401816095, 612.8432697749898, 320.8225009933049, 152.20441674637797, 121.93511823829118]
loss: 1220.566106579782
pde_losses: [36.21990044995337, 21.702528951123096]
bcs_losses: [1.7021324608557347, 7.992155837938316, 611.6127562770877, 287.35171034863686, 153.79878630224977, 100.18613595193713]
loss: 1196.810605531964
pde_losses: [27.345154875737446, 15.708076094328478]
bcs_losses: [0.1305108143903009, 4.715255839833738, 626.0965448750625, 270.0946550183383, 163.2834395358191, 89.43696847845422]
loss: 1186.1907794221966
pde_losses: [29.6807849090591, 16.273058822338854]
bcs_losses: [0.010270000428125196, 3.7385562866285325, 622.3973353921687, 263.8993621065972, 164.54342846799418, 85.647983436982]
loss: 1161.4592759138186
pde_losses: [73.21583537314069, 36.59416016345135]
bcs_losses: [0.8999854109213888, 1.7325580931137805, 566.3235948028157, 248.64484104469446, 157.4943254600327, 76.5539755656487]
loss: 1154.0629642518677
pde_losses: [64.65008880665589, 31.501325896488]
bcs_losses: [2.911775104428882, 0.8609421590269, 570.2313503275142, 239.9557683825571, 172.53716602953213, 71.41454754566443]
loss: 1150.5745764026883
pde_losses: [65.66294276275886, 31.552623589709494]
bcs_losses: [3.1442984260347124, 0.41287200218892, 572.0829040802635, 233.75600645351852, 176.1539087371463, 67.80902035106774]
loss: 1145.0756205457315
pde_losses: [67.78678170045515, 32.96773425431833]
bcs_losses: [3.008259540801172, 0.28049557527056257, 568.6270730170909, 230.93666483414103, 175.54811795238538, 65.92049367126913]
loss: 1131.6957620391986
pde_losses: [75.37608087287926, 36.91373326167276]
bcs_losses: [3.8026406605281924, 0.023542359780084718, 558.0857360209365, 219.73661525440315, 178.88939722820183, 58.86801638079689]
loss: 1117.6321725382882
pde_losses: [57.919930524666846, 32.034963058637395]
bcs_losses: [5.800082002080621, 0.1257196777100107, 566.8585905387469, 212.78932958523401, 188.40805188958925, 53.6955052616232]
loss: 1108.2471277485727
pde_losses: [53.082015558263194, 32.791407998019814]
bcs_losses: [6.312027374498278, 0.12281241144748767, 560.4373340156318, 213.24218422382904, 189.670401725194, 52.588944441689094]
loss: 1096.7391231014647
pde_losses: [53.699514285956305, 36.713838412816976]
bcs_losses: [6.278136686257087, 0.1548856738412741, 548.5031419518434, 213.53053419608403, 186.64855377380152, 51.21051812086422]
loss: 1082.183525203394
pde_losses: [48.639269373410166, 42.58136920988786]
bcs_losses: [6.74913416323334, 0.23264248258081277, 531.1692164522453, 217.8748973152663, 183.98314574637132, 50.95385046039878]
loss: 1071.2902964036825
pde_losses: [59.00821476635233, 52.29704747148019]
bcs_losses: [6.626678312836264, 0.2709874836124342, 509.7941215317487, 216.49985083024487, 177.5553886025242, 49.23800740488355]
loss: 1062.509146141326
pde_losses: [60.91308156085011, 53.78424675558512]
bcs_losses: [5.105353650490818, 0.3793391692622621, 501.9604784645144, 219.21839469237426, 171.81938703108773, 49.328864817161325]
loss: 1034.0722000656629
pde_losses: [79.7600411559233, 75.27620934781777]
bcs_losses: [2.438021238272824, 0.5296841187614831, 461.70808351155455, 216.39058269701667, 151.19541723246533, 46.77416076385076]
loss: 1006.4044957143242
pde_losses: [88.37065335996698, 103.0096835917969]
bcs_losses: [1.0021645828507848, 0.6632338873301298, 427.585279390784, 213.05237623808915, 127.61680739085908, 45.10429727264711]
loss: 992.2605175625907
pde_losses: [86.75965335085552, 115.67796933616701]
bcs_losses: [0.7934244344745675, 0.9436141378268426, 411.62595396433085, 212.96998114217695, 118.33504338927548, 45.154877807483544]
loss: 962.5572610305735
pde_losses: [79.64423851997029, 130.5085990184123]
bcs_losses: [0.8347950613123484, 2.5669838324068928, 387.2774811486391, 213.58626348984006, 101.6287888370824, 46.5101111229102]
loss: 944.7173340546061
pde_losses: [79.85851401114378, 130.96607674397595]
bcs_losses: [0.6668523900217107, 3.4204781787398257, 376.83836261389297, 209.26697034713368, 98.1400246530385, 45.5600551166597]
loss: 927.5016331532036
pde_losses: [82.91523011502223, 131.01215681578967]
bcs_losses: [0.4687323146228384, 4.6512293571614185, 363.68718081648166, 205.6822905659776, 94.077292246533, 45.007520921615]
loss: 895.4769704123357
pde_losses: [125.46752663967574, 140.36379860051548]
bcs_losses: [0.6798079113327716, 5.683594322019449, 314.4392636160069, 169.06280090360508, 103.94896282840632, 35.83121559077406]
loss: 862.5561950352379
pde_losses: [179.84837189778122, 127.46910996400871]
bcs_losses: [0.13853583424585997, 6.5643244290663105, 261.4576360836239, 147.93402843429996, 104.94854566977013, 34.19564272244178]
loss: 852.3722772709549
pde_losses: [203.51625618027495, 122.94285121026803]
bcs_losses: [0.299307214780615, 6.218721099667254, 242.34413183622615, 138.2165193472086, 106.25748835213327, 32.577002030396145]
loss: 843.9115217854862
pde_losses: [230.77987354465614, 120.01554939605542]
bcs_losses: [0.7814744371269573, 5.7668097123269915, 221.62661976615584, 131.51165375205076, 101.56017600226117, 31.869365174853048]
loss: 795.5826513952943
pde_losses: [284.00417112303904, 116.14223467592541]
bcs_losses: [6.646117994602807, 5.136300219769253, 157.88336453992724, 107.01263027614837, 88.17504018545273, 30.58279238042941]
loss: 790.9984204228341
pde_losses: [297.4741717658279, 120.68939398565932]
bcs_losses: [8.660254238822851, 5.167451725866114, 141.42134657905484, 101.41469487116966, 85.60273780042519, 30.56836945600819]
loss: 781.6248896984243
pde_losses: [303.0808791919287, 124.25658306694517]
bcs_losses: [15.628259737992742, 7.333156701280869, 117.0314050415058, 97.54276441436282, 79.227483042731, 37.524358501677334]
loss: 772.9982993322055
pde_losses: [303.6192045217246, 124.37069552195071]
bcs_losses: [16.083421629611404, 7.6509125580394475, 113.05690831826236, 93.00941853543125, 78.11860592644618, 37.08913232073951]
loss: 742.3306336077158
pde_losses: [305.4106360530854, 124.51411638985167]
bcs_losses: [16.07480715087176, 8.2369619305846, 100.24885486578985, 76.75061071451263, 77.02606629990902, 34.06858020311087]
loss: 733.3615488537037
pde_losses: [330.783858725017, 128.58418157338244]
bcs_losses: [15.348225260403433, 6.415886822924419, 82.6845497954761, 60.43821160241328, 79.0266379321719, 30.079997141915044]
loss: 715.7793890844557
pde_losses: [311.43249412593747, 141.2451188925038]
bcs_losses: [16.06991552508814, 6.833451684386989, 77.55511526547251, 53.62041178949412, 80.4525809762722, 28.570300825300478]
loss: 697.8636794958422
pde_losses: [287.4453746250428, 158.23837224006306]
bcs_losses: [16.660929438769912, 7.07093102000089, 73.03574490309362, 46.702058059063525, 82.20065180141822, 26.50961740839029]
loss: 673.8676447180017
pde_losses: [247.13997239695323, 193.45490755998975]
bcs_losses: [17.308547702040933, 8.681255393700905, 65.93262542639562, 35.210707013585754, 83.36983561674629, 22.769793608589232]
loss: 655.1603249015521
pde_losses: [209.13564887175548, 217.3484572694575]
bcs_losses: [17.79038410675217, 10.277295165889736, 62.73851372510252, 31.079610390726085, 83.90173430054361, 22.888681071325017]
loss: 640.0629211877994
pde_losses: [184.83174906425361, 229.25066023596113]
bcs_losses: [18.151307843007245, 11.269855663268759, 60.88189263149324, 28.743790025955782, 84.51599482516495, 22.417670898694645]
loss: 624.3056636890449
pde_losses: [157.50123631272385, 263.92742233411866]
bcs_losses: [18.603876336228936, 15.00624513967385, 53.14257079634316, 16.452986974839938, 85.78653907230239, 13.884786722814082]
loss: 618.2573561413842
pde_losses: [157.290714806418, 259.71506750227906]
bcs_losses: [17.89035951562547, 15.008092826713352, 52.60972676282138, 16.193041668594365, 85.76980633537694, 13.780546723555592]
loss: 521.9035329956456
pde_losses: [151.48832429580236, 182.41763985898967]
bcs_losses: [9.662277118719157, 10.350150568496847, 52.60909154359847, 13.072973195766567, 92.52104465211134, 9.782031762161159]
loss: 514.762270388964
pde_losses: [148.09312748565821, 178.62867723343905]
bcs_losses: [9.028359975074299, 9.88247886196338, 53.33267737876025, 12.963612036055112, 93.34485909825204, 9.488478319761692]
loss: 495.77851065312655
pde_losses: [130.90217143451216, 179.5573146770538]
bcs_losses: [4.905749543805953, 6.751013003417659, 53.2207323498656, 12.189119399937308, 99.60337421029551, 8.649036034238508]
loss: 472.239517942627
pde_losses: [97.87636062975629, 187.4617879893626]
bcs_losses: [1.3858982092278804, 2.911668380046697, 58.345687356364564, 10.062761896810356, 102.66111475243227, 11.53423872862636]
loss: 453.34447179299525
pde_losses: [92.3710011173222, 164.08250884490903]
bcs_losses: [1.3024677918751881, 2.5194962113824326, 62.01924481832521, 8.915117936701254, 105.31759011161957, 16.817044960860358]
loss: 439.45693357966525
pde_losses: [107.27908715613196, 134.8733811319364]
bcs_losses: [1.3714894876463566, 1.3560730131952214, 65.13404501996656, 5.769064784475045, 103.37717352429571, 20.29661946201794]
loss: 403.0147030424131
pde_losses: [85.69426323535474, 100.74121128255622]
bcs_losses: [7.59046683474006, 1.0079574357639431, 76.7499808345172, 2.155382059448983, 104.70965453137752, 24.365786828654425]
loss: 397.4898542929678
pde_losses: [83.50132920013489, 96.69843655101755]
bcs_losses: [8.068006516459521, 1.0332102863000734, 77.12187416396883, 1.9635443377802027, 104.46235302320406, 24.641100214102657]
loss: 384.01382478543866
pde_losses: [106.46536623538918, 62.93547406620759]
bcs_losses: [11.44208324858849, 1.7237977709368402, 76.0935426959567, 2.1226444846438004, 92.74284100447002, 30.488075279246047]
loss: 379.42814769556423
pde_losses: [103.16427333710477, 61.80653805999507]
bcs_losses: [11.441868757920608, 1.761079499411812, 75.99634366410348, 2.3803680745075497, 91.91466532389735, 30.96301097862358]
loss: 354.47460086822275
pde_losses: [84.20898003061151, 36.799904473350935]
bcs_losses: [12.707552425508617, 3.2365918913354546, 78.10716153002762, 10.01184801745044, 91.45879948597458, 37.94376301396357]
loss: 328.8641169529619
pde_losses: [62.64513581300705, 26.993885715718683]
bcs_losses: [13.392578500792407, 3.5667835501124276, 78.01464833224257, 16.266837234131632, 86.59056319271798, 41.393684614239206]
loss: 305.58359565292676
pde_losses: [46.570551367214435, 23.654243706486795]
bcs_losses: [12.660765847731135, 2.866799384522108, 78.15119122575261, 15.875314431209043, 86.5802629164705, 39.22446677354014]
loss: 294.76122480599315
pde_losses: [40.072424307671945, 19.385834582478616]
bcs_losses: [10.599289170142042, 2.065190683883531, 82.27462591354245, 16.73304402577937, 88.19884647324469, 35.431969649250526]
loss: 281.00462100700236
pde_losses: [56.393568692456064, 24.497485209177878]
bcs_losses: [5.783239529990375, 0.8362391405520423, 76.08953047327303, 11.323114145856255, 80.09820504668413, 25.983238769012555]
loss: 260.6133941833073
pde_losses: [59.32776945554445, 12.254073808428634]
bcs_losses: [2.2592764465789523, 0.7657894461750947, 70.09175345253675, 12.55317888812328, 74.56976785679902, 28.79178482912113]
loss: 237.02138059473702
pde_losses: [52.352515381662904, 19.970633773869213]
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loss: 226.30328739256532
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