Hamilton-Jacobi PDE Physics-Informed Neural Network (PINN) Loss Function Error vs Time Benchmarks
Adapted from NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations. Uses the NeuralPDE.jl library from the SciML Scientific Machine Learning Open Source Organization for the implementation of physics-informed neural networks (PINNs) and other science-guided AI techniques.
using NeuralPDE
using Integrals, Cubature, Cuba
using ModelingToolkit, Optimization, OptimizationOptimJL
using Lux, Plots
using OptimizationOptimisers
using DelimitedFiles
using QuasiMonteCarlo
import ModelingToolkit: Interval, infimum, supremumfunction hamilton_jacobi(strategy, minimizer, maxIters)
## DECLARATIONS
@parameters t x1 x2 x3 x4
@variables u(..)
Dt = Differential(t)
Dx1 = Differential(x1)
Dx2 = Differential(x2)
Dx3 = Differential(x3)
Dx4 = Differential(x4)
Dxx1 = Differential(x1)^2
Dxx2 = Differential(x2)^2
Dxx3 = Differential(x3)^2
Dxx4 = Differential(x4)^2
# Discretization
tmax = 1.0
x1width = 1.0
x2width = 1.0
x3width = 1.0
x4width = 1.0
tMeshNum = 10
x1MeshNum = 10
x2MeshNum = 10
x3MeshNum = 10
x4MeshNum = 10
dt = tmax / tMeshNum
dx1 = x1width / x1MeshNum
dx2 = x2width / x2MeshNum
dx3 = x3width / x3MeshNum
dx4 = x4width / x4MeshNum
domains = [t ∈ Interval(0.0, tmax),
x1 ∈ Interval(0.0, x1width),
x2 ∈ Interval(0.0, x2width),
x3 ∈ Interval(0.0, x3width),
x4 ∈ Interval(0.0, x4width)]
ts = 0.0:dt:tmax
x1s = 0.0:dx1:x1width
x2s = 0.0:dx2:x2width
x3s = 0.0:dx3:x3width
x4s = 0.0:dx4:x4width
λ = 1.0f0
# Operators
Δu = Dxx1(u(t, x1, x2, x3, x4)) + Dxx2(u(t, x1, x2, x3, x4)) +
Dxx3(u(t, x1, x2, x3, x4)) + Dxx4(u(t, x1, x2, x3, x4)) # Laplacian
∇u = [Dx1(u(t, x1, x2, x3, x4)), Dx2(u(t, x1, x2, x3, x4)),
Dx3(u(t, x1, x2, x3, x4)), Dx4(u(t, x1, x2, x3, x4))]
# Equation
eq = Dt(u(t, x1, x2, x3, x4)) + Δu - λ * sum(∇u .^ 2) ~ 0 #HAMILTON-JACOBI-BELLMAN EQUATION
terminalCondition = log((1 + x1 * x1 + x2 * x2 + x3 * x3 + x4 * x4) / 2) # see PNAS paper
bcs = [u(tmax, x1, x2, x3, x4) ~ terminalCondition] #PNAS paper again
## NEURAL NETWORK
n = 10 #neuron number
chain = Lux.Chain(Lux.Dense(5, n, tanh), Lux.Dense(n, n, tanh), Lux.Dense(n, 1)) #Neural network from OptimizationFlux library
indvars = [t, x1, x2, x3, x4] #physically independent variables
depvars = [u(t, x1, x2, x3, x4)] #dependent (target) variable
dim = length(domains)
losses = []
error = []
times = []
dx_err = 0.2
error_strategy = GridTraining(dx_err)
discretization_ = PhysicsInformedNN(chain, error_strategy)
@named pde_system_ = PDESystem(eq, bcs, domains, indvars, depvars)
prob_ = discretize(pde_system_, discretization_)
function loss_function_(θ, p)
params = θ.u
return prob_.f.f(params, nothing)
end
function cb_(p, l)
try
deltaT_s = time_ns()
ctime = time_ns() - startTime - timeCounter
push!(times, ctime / 1e9)
push!(losses, l)
# Extract parameters for loss calculation
params = p.u
loss_ = loss_function_(p, nothing)
push!(error, loss_)
timeCounter += time_ns() - deltaT_s
return false
catch e
@warn "Callback error: $e"
return false
end
end
@named pde_system = PDESystem(eq, bcs, domains, indvars, depvars)
discretization = NeuralPDE.PhysicsInformedNN(chain, strategy)
prob = NeuralPDE.discretize(pde_system, discretization)
timeCounter = 0.0
startTime = time_ns() #Fix initial time (t=0) before starting the training
res = Optimization.solve(prob, minimizer, callback = cb_, maxiters = maxIters)
phi = discretization.phi
params = res.minimizer
# Model prediction
domain = [ts, x1s, x2s, x3s, x4s]
u_predict = [reshape(
[first(phi([t, x1, x2, x3, x4], res.minimizer)) for x1 in x1s
for x2 in x2s for x3 in x3s for x4 in x4s],
(length(x1s), length(x2s), length(x3s), length(x4s))) for t in ts] #matrix of model's prediction
return [error, params, domain, times, losses]
end
maxIters = [(1, 1, 1, 1000, 1000, 1000, 1000), (1, 1, 1, 300, 300, 300, 300)] #iters for ADAM/LBFGS
# maxIters = [(1,1,1,1,1,2,2),(1,1,1,3,3,3,3)] #iters for ADAM/LBFGS
strategies = [
NeuralPDE.QuadratureTraining(quadrature_alg = CubaCuhre(), reltol = 1e-4, abstol = 1e-4, maxiters = 1100),
#NeuralPDE.QuadratureTraining(quadrature_alg = HCubatureJL(), reltol = 1e-4, abstol = 1e-4, maxiters = 1100, batch = 0),
NeuralPDE.GridTraining(0.1),
NeuralPDE.QuadratureTraining(quadrature_alg = CubatureJLh(), reltol = 1e-4, abstol = 1e-4, maxiters = 1100),
NeuralPDE.QuadratureTraining(quadrature_alg = CubatureJLp(), reltol = 1e-4, abstol = 1e-4, maxiters = 1100),
NeuralPDE.GridTraining(0.2),
NeuralPDE.StochasticTraining(400; bcs_points = 50),
NeuralPDE.QuasiRandomTraining(400; bcs_points = 50)]
strategies_short_name = ["CubaCuhre",
"HCubatureJL",
"CubatureJLh",
"CubatureJLp",
"GridTraining",
"StochasticTraining",
"QuasiRandomTraining"]
minimizers = [Optimisers.ADAM(0.005),
#BFGS()]
LBFGS()]
minimizers_short_name = ["ADAM",
"LBFGS"]
#"BFGS"]
# Run models
error_res = Dict()
domains = Dict()
params_res = Dict() #to use same params for the next run
times = Dict()
losses_res = Dict()Dict{Any, Any}()Solve
print("Starting run")
## Convergence
for min in 1:length(minimizers) # minimizer
for strat in 1:length(strategies) # strategy
# println(string(strategies_short_name[strat], " ", minimizers_short_name[min]))
res = hamilton_jacobi(strategies[strat], minimizers[min], maxIters[min][strat])
push!(error_res, string(strat, min) => res[1])
push!(params_res, string(strat, min) => res[2])
push!(domains, string(strat, min) => res[3])
push!(times, string(strat, min) => res[4])
push!(losses_res, string(strat, min) => res[5])
end
endStarting run#Plotting the first strategy with the first minimizer out from the loop to initialize the canvas
current_label = string(strategies_short_name[1], " + ", minimizers_short_name[1])
error = Plots.plot(times["11"], error_res["11"], yaxis = :log10, label = current_label)#, xlims = (0,10))#legend = true)#, size=(1200,700))
plot!(error, times["21"], error_res["21"], yaxis = :log10,
label = string(strategies_short_name[2], " + ", minimizers_short_name[1]))
plot!(error, times["31"], error_res["31"], yaxis = :log10,
label = string(strategies_short_name[3], " + ", minimizers_short_name[1]))
plot!(error, times["41"], error_res["41"], yaxis = :log10,
label = string(strategies_short_name[4], " + ", minimizers_short_name[1]))
plot!(error, times["51"], error_res["51"], yaxis = :log10,
label = string(strategies_short_name[5], " + ", minimizers_short_name[1]))
plot!(error, times["61"], error_res["61"], yaxis = :log10,
label = string(strategies_short_name[6], " + ", minimizers_short_name[1]))
plot!(error, times["71"], error_res["71"], yaxis = :log10,
label = string(strategies_short_name[7], " + ", minimizers_short_name[1]))
plot!(error, times["12"], error_res["12"], yaxis = :log10,
label = string(strategies_short_name[1], " + ", minimizers_short_name[2]))
plot!(error, times["22"], error_res["22"], yaxis = :log10,
label = string(strategies_short_name[2], " + ", minimizers_short_name[2]))
plot!(error, times["32"], error_res["32"], yaxis = :log10,
label = string(strategies_short_name[3], " + ", minimizers_short_name[2]))
plot!(error, times["42"], error_res["42"], yaxis = :log10,
label = string(strategies_short_name[4], " + ", minimizers_short_name[2]))
plot!(error, times["52"], error_res["52"], yaxis = :log10,
label = string(strategies_short_name[5], " + ", minimizers_short_name[2]))
plot!(error, times["62"], error_res["62"], yaxis = :log10,
label = string(strategies_short_name[6], " + ", minimizers_short_name[2]))
plot!(error, times["72"], error_res["72"], yaxis = :log10,
title = string("Hamilton Jacobi convergence ADAM/LBFGS"),
ylabel = "log(error)", xlabel = "t",
label = string(strategies_short_name[7], " + ", minimizers_short_name[2]))
Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/PINNErrorsVsTime","hamilton_jacobi_et.jmd")Computer Information:
Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNErrorsVsTime/Project.toml`
[8a292aeb] Cuba v2.3.0
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.And the full manifest:
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[b3c3ace0] RangeArrays v0.3.2
[c84ed2f1] Ratios v0.4.5
[c1ae055f] RealDot v0.1.0
[3cdcf5f2] RecipesBase v1.3.4
[01d81517] RecipesPipeline v0.6.12
⌃ [731186ca] RecursiveArrayTools v3.31.0
[189a3867] Reexport v1.2.2
[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
[ae5879a3] ResettableStacks v1.1.1
⌃ [37e2e3b7] ReverseDiff v1.15.3
⌅ [79098fc4] Rmath v0.8.0
⌃ [7e49a35a] RuntimeGeneratedFunctions v0.5.13
⌃ [9dfe8606] SCCNonlinearSolve v1.0.0
[94e857df] SIMDTypes v0.1.0
[476501e8] SLEEFPirates v0.6.43
⌃ [0bca4576] SciMLBase v2.75.1
[31c91b34] SciMLBenchmarks v0.1.3
⌃ [19f34311] SciMLJacobianOperators v0.1.1
⌅ [c0aeaf25] SciMLOperators v0.3.12
[53ae85a6] SciMLStructures v1.7.0
[30f210dd] ScientificTypesBase v3.0.0
⌃ [7e506255] ScopedValues v1.3.0
⌃ [6c6a2e73] Scratch v1.2.1
[efcf1570] Setfield v1.1.2
[992d4aef] Showoff v1.0.3
[777ac1f9] SimpleBufferStream v1.2.0
⌃ [727e6d20] SimpleNonlinearSolve v2.1.0
⌃ [699a6c99] SimpleTraits v0.9.4
[ce78b400] SimpleUnPack v1.1.0
[ed01d8cd] Sobol v1.5.0
[b85f4697] SoftGlobalScope v1.1.0
⌃ [a2af1166] SortingAlgorithms v1.2.1
⌅ [9f842d2f] SparseConnectivityTracer v0.6.13
[dc90abb0] SparseInverseSubset v0.1.2
⌃ [0a514795] SparseMatrixColorings v0.4.14
⌃ [276daf66] SpecialFunctions v2.5.0
[171d559e] SplittablesBase v0.1.15
⌃ [860ef19b] StableRNGs v1.0.2
⌃ [aedffcd0] Static v1.1.1
[0d7ed370] StaticArrayInterface v1.8.0
⌃ [90137ffa] StaticArrays v1.9.13
[1e83bf80] StaticArraysCore v1.4.3
⌃ [64bff920] StatisticalTraits v3.4.0
⌃ [82ae8749] StatsAPI v1.7.0
⌃ [2913bbd2] StatsBase v0.34.4
⌃ [4c63d2b9] StatsFuns v1.3.2
⌃ [7792a7ef] StrideArraysCore v0.5.7
[69024149] StringEncodings v0.3.7
[892a3eda] StringManipulation v0.4.1
⌃ [09ab397b] StructArrays v0.7.0
⌃ [2efcf032] SymbolicIndexingInterface v0.3.38
⌃ [19f23fe9] SymbolicLimits v0.2.2
⌅ [d1185830] SymbolicUtils v3.17.0
⌃ [0c5d862f] Symbolics v6.30.0
[3783bdb8] TableTraits v1.0.1
⌃ [bd369af6] Tables v1.12.0
⌃ [ed4db957] TaskLocalValues v0.1.2
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.0
[5d786b92] TerminalLoggers v0.1.7
[1c621080] TestItems v1.0.0
⌃ [8290d209] ThreadingUtilities v0.5.2
⌃ [a759f4b9] TimerOutputs v0.5.28
[0796e94c] Tokenize v0.5.29
[3bb67fe8] TranscodingStreams v0.11.3
⌃ [28d57a85] Transducers v0.4.84
⌃ [410a4b4d] Tricks v0.1.10
[781d530d] TruncatedStacktraces v1.4.0
⌃ [5c2747f8] URIs v1.5.1
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
⌅ [1986cc42] Unitful v1.22.0
⌃ [45397f5d] UnitfulLatexify v1.6.4
[a7c27f48] Unityper v0.1.6
[013be700] UnsafeAtomics v0.3.0
[41fe7b60] Unzip v0.2.0
⌃ [3d5dd08c] VectorizationBase v0.21.71
[81def892] VersionParsing v1.3.0
[897b6980] WeakValueDicts v0.1.0
[44d3d7a6] Weave v0.10.12
⌃ [d49dbf32] WeightInitializers v1.1.1
[efce3f68] WoodburyMatrices v1.0.0
⌃ [ddb6d928] YAML v0.4.13
⌃ [c2297ded] ZMQ v1.4.0
⌃ [e88e6eb3] Zygote v0.6.75
[700de1a5] ZygoteRules v0.2.7
[6e34b625] Bzip2_jll v1.0.9+0
⌃ [83423d85] Cairo_jll v1.18.2+1
[3bed1096] Cuba_jll v4.2.2+1
[7bc98958] Cubature_jll v1.0.5+0
⌃ [ee1fde0b] Dbus_jll v1.14.10+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
⌃ [2e619515] Expat_jll v2.6.5+0
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
⌃ [f5851436] FFTW_jll v3.3.10+3
⌃ [a3f928ae] Fontconfig_jll v2.15.0+0
⌃ [d7e528f0] FreeType2_jll v2.13.3+1
⌃ [559328eb] FriBidi_jll v1.0.16+0
[0656b61e] GLFW_jll v3.4.0+2
⌅ [d2c73de3] GR_jll v0.73.13+0
[78b55507] Gettext_jll v0.21.0+0
⌃ [f8c6e375] Git_jll v2.47.1+0
⌃ [7746bdde] Glib_jll v2.82.4+0
⌃ [3b182d85] Graphite2_jll v1.3.14+1
⌃ [2e76f6c2] HarfBuzz_jll v8.5.0+0
⌃ [e33a78d0] Hwloc_jll v2.12.0+0
⌃ [1d5cc7b8] IntelOpenMP_jll v2025.0.4+0
⌃ [aacddb02] JpegTurbo_jll v3.1.1+0
⌃ [c1c5ebd0] LAME_jll v3.100.2+0
[88015f11] LERC_jll v4.0.1+0
⌅ [dad2f222] LLVMExtra_jll v0.0.35+0
⌃ [1d63c593] LLVMOpenMP_jll v18.1.7+0
[dd4b983a] LZO_jll v2.10.3+0
[81d17ec3] L_BFGS_B_jll v3.0.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+2
⌃ [d4300ac3] Libgcrypt_jll v1.11.0+0
⌃ [7e76a0d4] Libglvnd_jll v1.7.0+0
[7add5ba3] Libgpg_error_jll v1.51.1+0
[94ce4f54] Libiconv_jll v1.18.0+0
⌃ [4b2f31a3] Libmount_jll v2.40.3+0
⌃ [89763e89] Libtiff_jll v4.7.1+0
⌃ [38a345b3] Libuuid_jll v2.40.3+0
⌃ [856f044c] MKL_jll v2025.0.1+1
⌃ [e7412a2a] Ogg_jll v1.3.5+1
⌃ [458c3c95] OpenSSL_jll v3.0.16+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
⌃ [91d4177d] Opus_jll v1.3.3+0
⌃ [36c8627f] Pango_jll v1.56.1+0
⌅ [30392449] Pixman_jll v0.43.4+0
⌅ [c0090381] Qt6Base_jll v6.7.1+1
⌅ [629bc702] Qt6Declarative_jll v6.7.1+2
⌅ [ce943373] Qt6ShaderTools_jll v6.7.1+1
⌃ [e99dba38] Qt6Wayland_jll v6.7.1+1
[f50d1b31] Rmath_jll v0.5.1+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
⌃ [a2964d1f] Wayland_jll v1.21.0+2
⌃ [2381bf8a] Wayland_protocols_jll v1.36.0+0
⌅ [02c8fc9c] XML2_jll v2.13.6+1
⌃ [aed1982a] XSLT_jll v1.1.42+0
⌃ [ffd25f8a] XZ_jll v5.6.4+1
⌃ [f67eecfb] Xorg_libICE_jll v1.1.1+0
⌃ [c834827a] Xorg_libSM_jll v1.2.4+0
⌃ [4f6342f7] Xorg_libX11_jll v1.8.6+3
⌃ [0c0b7dd1] Xorg_libXau_jll v1.0.12+0
⌃ [935fb764] Xorg_libXcursor_jll v1.2.3+0
⌃ [a3789734] Xorg_libXdmcp_jll v1.1.5+0
⌃ [1082639a] Xorg_libXext_jll v1.3.6+3
⌃ [d091e8ba] Xorg_libXfixes_jll v6.0.0+0
⌃ [a51aa0fd] Xorg_libXi_jll v1.8.2+0
⌃ [d1454406] Xorg_libXinerama_jll v1.1.5+0
⌃ [ec84b674] Xorg_libXrandr_jll v1.5.4+0
⌃ [ea2f1a96] Xorg_libXrender_jll v0.9.11+1
⌃ [14d82f49] Xorg_libpthread_stubs_jll v0.1.2+0
⌃ [c7cfdc94] Xorg_libxcb_jll v1.17.0+3
⌃ [cc61e674] Xorg_libxkbfile_jll v1.1.2+1
⌃ [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
⌃ [12413925] Xorg_xcb_util_image_jll v0.4.0+1
⌃ [2def613f] Xorg_xcb_util_jll v0.4.0+1
⌃ [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
⌃ [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
⌃ [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
⌃ [35661453] Xorg_xkbcomp_jll v1.4.6+1
⌃ [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
⌃ [c5fb5394] Xorg_xtrans_jll v1.5.1+0
⌃ [8f1865be] ZeroMQ_jll v4.3.5+3
[3161d3a3] Zstd_jll v1.5.7+1
⌃ [35ca27e7] eudev_jll v3.2.9+0
⌅ [214eeab7] fzf_jll v0.56.3+0
⌃ [1a1c6b14] gperf_jll v3.1.1+1
⌃ [a4ae2306] libaom_jll v3.11.0+0
⌅ [0ac62f75] libass_jll v0.15.2+0
[1183f4f0] libdecor_jll v0.2.2+0
⌃ [2db6ffa8] libevdev_jll v1.11.0+0
⌃ [f638f0a6] libfdk_aac_jll v2.0.3+0
⌃ [36db933b] libinput_jll v1.18.0+0
⌃ [b53b4c65] libpng_jll v1.6.47+0
⌃ [a9144af2] libsodium_jll v1.0.20+3
⌃ [f27f6e37] libvorbis_jll v1.3.7+2
⌃ [009596ad] mtdev_jll v1.1.6+0
⌃ [1317d2d5] oneTBB_jll v2022.0.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
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⌃ [d8fb68d0] xkbcommon_jll v1.4.1+2
[0dad84c5] ArgTools v1.1.1
[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8ba89e20] Distributed
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.10.0
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays v1.10.0
[10745b16] Statistics v1.10.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.1.1+0
[deac9b47] LibCURL_jll v8.4.0+0
[e37daf67] LibGit2_jll v1.6.4+0
[29816b5a] LibSSH2_jll v1.11.0+1
[c8ffd9c3] MbedTLS_jll v2.28.2+1
[14a3606d] MozillaCACerts_jll v2023.1.10
[4536629a] OpenBLAS_jll v0.3.23+4
[05823500] OpenLibm_jll v0.8.1+2
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.2.1+1
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[8e850ede] nghttp2_jll v1.52.0+1
[3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.