diff --git a/previews/PR249/.documenter-siteinfo.json b/previews/PR249/.documenter-siteinfo.json
index b13f688..afa563e 100644
--- a/previews/PR249/.documenter-siteinfo.json
+++ b/previews/PR249/.documenter-siteinfo.json
@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.1","generation_timestamp":"2024-02-18T11:26:09","documenter_version":"1.2.1"}}
\ No newline at end of file
+{"documenter":{"julia_version":"1.10.1","generation_timestamp":"2024-02-26T17:52:05","documenter_version":"1.2.1"}}
\ No newline at end of file
diff --git a/previews/PR249/assets/Manifest.toml b/previews/PR249/assets/Manifest.toml
index bbcbcdc..99d2846 100644
--- a/previews/PR249/assets/Manifest.toml
+++ b/previews/PR249/assets/Manifest.toml
@@ -2,7 +2,7 @@
julia_version = "1.10.1"
manifest_format = "2.0"
-project_hash = "adfa74819f53813ac8a6bfd6d95095d6b0655cea"
+project_hash = "de0d6e45a22e6a4510dc57906c315a0b297eefd6"
[[deps.ADTypes]]
git-tree-sha1 = "41c37aa88889c171f1300ceac1313c06e891d245"
@@ -14,12 +14,6 @@ git-tree-sha1 = "574baf8110975760d391c710b6341da1afa48d8c"
uuid = "a4c015fc-c6ff-483c-b24f-f7ea428134e9"
version = "0.0.1"
-[[deps.ASL_jll]]
-deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
-git-tree-sha1 = "6252039f98492252f9e47c312c8ffda0e3b9e78d"
-uuid = "ae81ac8f-d209-56e5-92de-9978fef736f9"
-version = "0.1.3+0"
-
[[deps.AbstractFFTs]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef"
@@ -32,15 +26,15 @@ weakdeps = ["ChainRulesCore", "Test"]
AbstractFFTsTestExt = "Test"
[[deps.AbstractTrees]]
-git-tree-sha1 = "faa260e4cb5aba097a73fab382dd4b5819d8ec8c"
+git-tree-sha1 = "2d9c9a55f9c93e8887ad391fbae72f8ef55e1177"
uuid = "1520ce14-60c1-5f80-bbc7-55ef81b5835c"
-version = "0.4.4"
+version = "0.4.5"
[[deps.Adapt]]
deps = ["LinearAlgebra", "Requires"]
-git-tree-sha1 = "cde29ddf7e5726c9fb511f340244ea3481267608"
+git-tree-sha1 = "0fb305e0253fd4e833d486914367a2ee2c2e78d0"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
-version = "3.7.2"
+version = "4.0.1"
weakdeps = ["StaticArrays"]
[deps.Adapt.extensions]
@@ -144,9 +138,9 @@ weakdeps = ["HTTP", "Sockets"]
[[deps.BoundaryValueDiffEq]]
deps = ["ADTypes", "Adapt", "ArrayInterface", "BandedMatrices", "ConcreteStructs", "DiffEqBase", "FastAlmostBandedMatrices", "ForwardDiff", "LinearAlgebra", "LinearSolve", "NonlinearSolve", "PreallocationTools", "PrecompileTools", "Preferences", "RecursiveArrayTools", "Reexport", "SciMLBase", "Setfield", "SparseArrays", "SparseDiffTools", "Tricks", "TruncatedStacktraces", "UnPack"]
-git-tree-sha1 = "dd234c9a030350d5ff4c45761d6cad0cfb358cb9"
+git-tree-sha1 = "3ff968887be48760b0e9e8650c2d05c96cdea9d8"
uuid = "764a87c0-6b3e-53db-9096-fe964310641d"
-version = "5.6.0"
+version = "5.6.3"
[deps.BoundaryValueDiffEq.extensions]
BoundaryValueDiffEqODEInterfaceExt = "ODEInterface"
@@ -209,9 +203,9 @@ version = "1.63.0"
[[deps.ChainRulesCore]]
deps = ["Compat", "LinearAlgebra"]
-git-tree-sha1 = "ad25e7d21ce10e01de973cdc68ad0f850a953c52"
+git-tree-sha1 = "aef70bb349b20aa81a82a19704c3ef339d4ee494"
uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
-version = "1.21.1"
+version = "1.22.1"
weakdeps = ["SparseArrays"]
[deps.ChainRulesCore.extensions]
@@ -287,9 +281,9 @@ version = "0.3.0"
[[deps.Compat]]
deps = ["TOML", "UUIDs"]
-git-tree-sha1 = "75bd5b6fc5089df449b5d35fa501c846c9b6549b"
+git-tree-sha1 = "c955881e3c981181362ae4088b35995446298b80"
uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
-version = "4.12.0"
+version = "4.14.0"
weakdeps = ["Dates", "LinearAlgebra"]
[deps.Compat.extensions]
@@ -356,9 +350,9 @@ version = "1.16.0"
[[deps.DataStructures]]
deps = ["Compat", "InteractiveUtils", "OrderedCollections"]
-git-tree-sha1 = "ac67408d9ddf207de5cfa9a97e114352430f01ed"
+git-tree-sha1 = "1fb174f0d48fe7d142e1109a10636bc1d14f5ac2"
uuid = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
-version = "0.18.16"
+version = "0.18.17"
[[deps.DataValueInterfaces]]
git-tree-sha1 = "bfc1187b79289637fa0ef6d4436ebdfe6905cbd6"
@@ -371,9 +365,9 @@ uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"
[[deps.DelayDiffEq]]
deps = ["ArrayInterface", "DataStructures", "DiffEqBase", "LinearAlgebra", "Logging", "OrdinaryDiffEq", "Printf", "RecursiveArrayTools", "Reexport", "SciMLBase", "SimpleNonlinearSolve", "SimpleUnPack"]
-git-tree-sha1 = "6725c56e3e3d563e37d8fd5e6c5eb66ac19321fd"
+git-tree-sha1 = "10c892dd12113644bd6c7fae520691f8a78032dc"
uuid = "bcd4f6db-9728-5f36-b5f7-82caef46ccdb"
-version = "5.46.0"
+version = "5.47.0"
[[deps.DelimitedFiles]]
deps = ["Mmap"]
@@ -395,9 +389,9 @@ version = "0.1.0+0"
[[deps.DiffEqBase]]
deps = ["ArrayInterface", "DataStructures", "DocStringExtensions", "EnumX", "EnzymeCore", "FastBroadcast", "ForwardDiff", "FunctionWrappers", "FunctionWrappersWrappers", "LinearAlgebra", "Logging", "Markdown", "MuladdMacro", "Parameters", "PreallocationTools", "PrecompileTools", "Printf", "RecursiveArrayTools", "Reexport", "SciMLBase", "SciMLOperators", "Setfield", "SparseArrays", "Static", "StaticArraysCore", "Statistics", "Tricks", "TruncatedStacktraces"]
-git-tree-sha1 = "3089c8295ab6d7c728cd6929121c1b4567457306"
+git-tree-sha1 = "2ad3a2dcd5f28f535aa884d199cc2f0a9d335729"
uuid = "2b5f629d-d688-5b77-993f-72d75c75574e"
-version = "6.147.0"
+version = "6.147.1"
[deps.DiffEqBase.extensions]
DiffEqBaseChainRulesCoreExt = "ChainRulesCore"
@@ -543,9 +537,9 @@ version = "1.0.4"
[[deps.Enzyme]]
deps = ["CEnum", "EnzymeCore", "Enzyme_jll", "GPUCompiler", "LLVM", "Libdl", "LinearAlgebra", "ObjectFile", "Preferences", "Printf", "Random"]
-git-tree-sha1 = "090a6f1996d357329bea66f0b1c7621b64d70bb0"
+git-tree-sha1 = "16b3a7980cdbf6f036ab46f7e9131afb273f1231"
uuid = "7da242da-08ed-463a-9acd-ee780be4f1d9"
-version = "0.11.16"
+version = "0.11.17"
weakdeps = ["SpecialFunctions"]
[deps.Enzyme.extensions]
@@ -736,15 +730,15 @@ version = "3.3.9+0"
[[deps.GPUArrays]]
deps = ["Adapt", "GPUArraysCore", "LLVM", "LinearAlgebra", "Printf", "Random", "Reexport", "Serialization", "Statistics"]
-git-tree-sha1 = "85d7fb51afb3def5dcb85ad31c3707795c8bccc1"
+git-tree-sha1 = "47e4686ec18a9620850bad110b79966132f14283"
uuid = "0c68f7d7-f131-5f86-a1c3-88cf8149b2d7"
-version = "9.1.0"
+version = "10.0.2"
[[deps.GPUArraysCore]]
deps = ["Adapt"]
-git-tree-sha1 = "2d6ca471a6c7b536127afccfa7564b5b39227fe0"
+git-tree-sha1 = "ec632f177c0d990e64d955ccc1b8c04c485a0950"
uuid = "46192b85-c4d5-4398-a991-12ede77f4527"
-version = "0.1.5"
+version = "0.1.6"
[[deps.GPUCompiler]]
deps = ["ExprTools", "InteractiveUtils", "LLVM", "Libdl", "Logging", "Scratch", "TimerOutputs", "UUIDs"]
@@ -778,15 +772,15 @@ version = "0.21.0+0"
[[deps.Git]]
deps = ["Git_jll"]
-git-tree-sha1 = "51764e6c2e84c37055e846c516e9015b4a291c7d"
+git-tree-sha1 = "04eff47b1354d702c3a85e8ab23d539bb7d5957e"
uuid = "d7ba0133-e1db-5d97-8f8c-041e4b3a1eb2"
-version = "1.3.0"
+version = "1.3.1"
[[deps.Git_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "LibCURL_jll", "Libdl", "Libiconv_jll", "OpenSSL_jll", "PCRE2_jll", "Zlib_jll"]
-git-tree-sha1 = "b30c473c97fcc1e1e44fab8f3e88fd1b89c9e9d1"
+git-tree-sha1 = "99f27817475017260f1ff24c771a5efef5765e34"
uuid = "f8c6e375-362e-5223-8a59-34ff63f689eb"
-version = "2.43.0+0"
+version = "2.44.0+0"
[[deps.Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE2_jll", "Zlib_jll"]
@@ -883,12 +877,6 @@ weakdeps = ["Random", "RecipesBase", "Statistics"]
IntervalSetsRecipesBaseExt = "RecipesBase"
IntervalSetsStatisticsExt = "Statistics"
-[[deps.Ipopt_jll]]
-deps = ["ASL_jll", "Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "MUMPS_seq_jll", "OpenBLAS32_jll", "Pkg"]
-git-tree-sha1 = "e3e202237d93f18856b6ff1016166b0f172a49a8"
-uuid = "9cc047cb-c261-5740-88fc-0cf96f7bdcc7"
-version = "300.1400.400+0"
-
[[deps.IrrationalConstants]]
git-tree-sha1 = "630b497eafcc20001bba38a4651b327dcfc491d2"
uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
@@ -919,15 +907,15 @@ version = "0.21.4"
[[deps.JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
-git-tree-sha1 = "60b1194df0a3298f460063de985eae7b01bc011a"
+git-tree-sha1 = "3336abae9a713d2210bb57ab484b1e065edd7d23"
uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8"
-version = "3.0.1+0"
+version = "3.0.2+0"
[[deps.JuliaFormatter]]
deps = ["CSTParser", "Combinatorics", "CommonMark", "DataStructures", "Glob", "Pkg", "PrecompileTools", "Tokenize"]
-git-tree-sha1 = "40f1eb7837b7b75d07c062790942406b9a010725"
+git-tree-sha1 = "fde717f9e3fe6ffb0336bcc2142cfa71dd9df1f5"
uuid = "98e50ef6-434e-11e9-1051-2b60c6c9e899"
-version = "1.0.49"
+version = "1.0.50"
[[deps.JumpProcesses]]
deps = ["ArrayInterface", "DataStructures", "DiffEqBase", "DocStringExtensions", "FunctionWrappers", "Graphs", "LinearAlgebra", "Markdown", "PoissonRandom", "Random", "RandomNumbers", "RecursiveArrayTools", "Reexport", "SciMLBase", "StaticArrays", "UnPack"]
@@ -941,15 +929,15 @@ weakdeps = ["FastBroadcast"]
[[deps.KLU]]
deps = ["LinearAlgebra", "SparseArrays", "SuiteSparse_jll"]
-git-tree-sha1 = "884c2968c2e8e7e6bf5956af88cb46aa745c854b"
+git-tree-sha1 = "01bb1d420d959b9f668e252a5732444a61ff25de"
uuid = "ef3ab10e-7fda-4108-b977-705223b18434"
-version = "0.4.1"
+version = "0.5.0"
[[deps.KernelAbstractions]]
deps = ["Adapt", "Atomix", "InteractiveUtils", "LinearAlgebra", "MacroTools", "PrecompileTools", "Requires", "SparseArrays", "StaticArrays", "UUIDs", "UnsafeAtomics", "UnsafeAtomicsLLVM"]
-git-tree-sha1 = "4e0cb2f5aad44dcfdc91088e85dee4ecb22c791c"
+git-tree-sha1 = "c7753cc3febe006708ce6798482004241f7d890b"
uuid = "63c18a36-062a-441e-b654-da1e3ab1ce7c"
-version = "0.9.16"
+version = "0.9.17"
weakdeps = ["EnzymeCore"]
[deps.KernelAbstractions.extensions]
@@ -975,9 +963,9 @@ version = "3.0.0+1"
[[deps.LLVM]]
deps = ["CEnum", "LLVMExtra_jll", "Libdl", "Preferences", "Printf", "Requires", "Unicode"]
-git-tree-sha1 = "9e70165cca7459d25406367f0c55e517a9a7bfe7"
+git-tree-sha1 = "ddab4d40513bce53c8e3157825e245224f74fae7"
uuid = "929cbde3-209d-540e-8aea-75f648917ca0"
-version = "6.5.0"
+version = "6.6.0"
[deps.LLVM.extensions]
BFloat16sExt = "BFloat16s"
@@ -987,9 +975,9 @@ version = "6.5.0"
[[deps.LLVMExtra_jll]]
deps = ["Artifacts", "JLLWrappers", "LazyArtifacts", "Libdl", "TOML"]
-git-tree-sha1 = "114e3a48f13d4c18ddd7fd6a00107b4b96f60f9c"
+git-tree-sha1 = "88b916503aac4fb7f701bb625cd84ca5dd1677bc"
uuid = "dad2f222-ce93-54a1-a47d-0025e8a3acab"
-version = "0.0.28+0"
+version = "0.0.29+0"
[[deps.LLVMOpenMP_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
@@ -1168,10 +1156,10 @@ deps = ["Libdl", "OpenBLAS_jll", "libblastrampoline_jll"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
[[deps.LinearSolve]]
-deps = ["ArrayInterface", "ConcreteStructs", "DocStringExtensions", "EnumX", "FastLapackInterface", "GPUArraysCore", "InteractiveUtils", "KLU", "Krylov", "Libdl", "LinearAlgebra", "MKL_jll", "PrecompileTools", "Preferences", "RecursiveFactorization", "Reexport", "SciMLBase", "SciMLOperators", "Setfield", "SparseArrays", "Sparspak", "StaticArraysCore", "UnPack"]
-git-tree-sha1 = "6f8e084deabe3189416c4e505b1c53e1b590cae8"
+deps = ["ArrayInterface", "ChainRulesCore", "ConcreteStructs", "DocStringExtensions", "EnumX", "FastLapackInterface", "GPUArraysCore", "InteractiveUtils", "KLU", "Krylov", "Libdl", "LinearAlgebra", "MKL_jll", "Markdown", "PrecompileTools", "Preferences", "RecursiveFactorization", "Reexport", "SciMLBase", "SciMLOperators", "Setfield", "SparseArrays", "Sparspak", "StaticArraysCore", "UnPack"]
+git-tree-sha1 = "40e698d20063b1a8dbbe40b259a24f7a71f53298"
uuid = "7ed4a6bd-45f5-4d41-b270-4a48e9bafcae"
-version = "2.22.1"
+version = "2.25.0"
[deps.LinearSolve.extensions]
LinearSolveBandedMatricesExt = "BandedMatrices"
@@ -1238,12 +1226,6 @@ weakdeps = ["ChainRulesCore", "ForwardDiff", "SpecialFunctions"]
ForwardDiffExt = ["ChainRulesCore", "ForwardDiff"]
SpecialFunctionsExt = "SpecialFunctions"
-[[deps.METIS_jll]]
-deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
-git-tree-sha1 = "1fd0a97409e418b78c53fac671cf4622efdf0f21"
-uuid = "d00139f3-1899-568f-a2f0-47f597d42d70"
-version = "5.1.2+0"
-
[[deps.MKL_jll]]
deps = ["Artifacts", "IntelOpenMP_jll", "JLLWrappers", "LazyArtifacts", "Libdl"]
git-tree-sha1 = "72dc3cf284559eb8f53aa593fe62cb33f83ed0c0"
@@ -1255,12 +1237,6 @@ git-tree-sha1 = "bc38dff0548128765760c79eb7388a4b37fae2c8"
uuid = "d8e11817-5142-5d16-987a-aa16d5891078"
version = "0.4.17"
-[[deps.MUMPS_seq_jll]]
-deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "METIS_jll", "OpenBLAS32_jll", "Pkg"]
-git-tree-sha1 = "29de2841fa5aefe615dea179fcde48bb87b58f57"
-uuid = "d7ed1dd3-d0ae-5e8e-bfb4-87a502085b8d"
-version = "5.4.1+0"
-
[[deps.MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "2fa9ee3e63fd3a4f7a9a4f4744a52f4856de82df"
@@ -1284,9 +1260,9 @@ version = "0.1.2"
[[deps.MathOptInterface]]
deps = ["BenchmarkTools", "CodecBzip2", "CodecZlib", "DataStructures", "ForwardDiff", "JSON", "LinearAlgebra", "MutableArithmetics", "NaNMath", "OrderedCollections", "PrecompileTools", "Printf", "SparseArrays", "SpecialFunctions", "Test", "Unicode"]
-git-tree-sha1 = "569a003f93d7c64068d3afaab908d21f67a22cd5"
+git-tree-sha1 = "e8b98c868029d007102dc5f98986c81f33b0ec37"
uuid = "b8f27783-ece8-5eb3-8dc8-9495eed66fee"
-version = "1.25.3"
+version = "1.26.0"
[[deps.MatrixFactorizations]]
deps = ["ArrayLayouts", "LinearAlgebra", "Printf", "Random"]
@@ -1418,9 +1394,9 @@ version = "1.2.0"
[[deps.NonlinearSolve]]
deps = ["ADTypes", "ArrayInterface", "ConcreteStructs", "DiffEqBase", "FastBroadcast", "FastClosures", "FiniteDiff", "ForwardDiff", "LazyArrays", "LineSearches", "LinearAlgebra", "LinearSolve", "MaybeInplace", "PrecompileTools", "Preferences", "Printf", "RecursiveArrayTools", "Reexport", "SciMLBase", "SimpleNonlinearSolve", "SparseArrays", "SparseDiffTools", "StaticArraysCore", "TimerOutputs"]
-git-tree-sha1 = "b377521f4810057a99b0fa8cb7a1311c6cb1c8cd"
+git-tree-sha1 = "f409959245f04c8004bd3711915d71c93b2043f7"
uuid = "8913a72c-1f9b-4ce2-8d82-65094dcecaec"
-version = "3.5.6"
+version = "3.7.3"
[deps.NonlinearSolve.extensions]
NonlinearSolveBandedMatricesExt = "BandedMatrices"
@@ -1428,6 +1404,7 @@ version = "3.5.6"
NonlinearSolveFixedPointAccelerationExt = "FixedPointAcceleration"
NonlinearSolveLeastSquaresOptimExt = "LeastSquaresOptim"
NonlinearSolveMINPACKExt = "MINPACK"
+ NonlinearSolveNLSolversExt = "NLSolvers"
NonlinearSolveNLsolveExt = "NLsolve"
NonlinearSolveSIAMFANLEquationsExt = "SIAMFANLEquations"
NonlinearSolveSpeedMappingExt = "SpeedMapping"
@@ -1440,6 +1417,7 @@ version = "3.5.6"
FixedPointAcceleration = "817d07cb-a79a-5c30-9a31-890123675176"
LeastSquaresOptim = "0fc2ff8b-aaa3-5acd-a817-1944a5e08891"
MINPACK = "4854310b-de5a-5eb6-a2a5-c1dee2bd17f9"
+ NLSolvers = "337daf1e-9722-11e9-073e-8b9effe078ba"
NLsolve = "2774e3e8-f4cf-5e23-947b-6d7e65073b56"
SIAMFANLEquations = "084e46ad-d928-497d-ad5e-07fa361a48c4"
SpeedMapping = "f1835b91-879b-4a3f-a438-e4baacf14412"
@@ -1467,12 +1445,6 @@ git-tree-sha1 = "887579a3eb005446d514ab7aeac5d1d027658b8f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+1"
-[[deps.OpenBLAS32_jll]]
-deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl"]
-git-tree-sha1 = "6065c4cff8fee6c6770b277af45d5082baacdba1"
-uuid = "656ef2d0-ae68-5445-9ca0-591084a874a2"
-version = "0.3.24+0"
-
[[deps.OpenBLAS_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
@@ -1537,10 +1509,10 @@ uuid = "3e6eede4-6085-4f62-9a71-46d9bc1eb92b"
version = "0.2.1"
[[deps.OptimizationMOI]]
-deps = ["Ipopt_jll", "MathOptInterface", "ModelingToolkit", "Optimization", "Reexport", "SparseArrays", "SymbolicIndexingInterface", "Symbolics"]
-git-tree-sha1 = "570e686c7a076ce2439eafb6d75499697a9dbfdf"
+deps = ["MathOptInterface", "ModelingToolkit", "Optimization", "Reexport", "SparseArrays", "SymbolicIndexingInterface", "Symbolics"]
+git-tree-sha1 = "75c381f438afd5c474f460442af6738d6d597b76"
uuid = "fd9f6733-72f4-499f-8506-86b2bdd0dea1"
-version = "0.3.4"
+version = "0.4.0"
[[deps.OptimizationNLopt]]
deps = ["NLopt", "Optimization", "Reexport"]
@@ -1567,9 +1539,9 @@ version = "1.6.3"
[[deps.OrdinaryDiffEq]]
deps = ["ADTypes", "Adapt", "ArrayInterface", "DataStructures", "DiffEqBase", "DocStringExtensions", "ExponentialUtilities", "FastBroadcast", "FastClosures", "FillArrays", "FiniteDiff", "ForwardDiff", "FunctionWrappersWrappers", "IfElse", "InteractiveUtils", "LineSearches", "LinearAlgebra", "LinearSolve", "Logging", "MacroTools", "MuladdMacro", "NonlinearSolve", "Polyester", "PreallocationTools", "PrecompileTools", "Preferences", "RecursiveArrayTools", "Reexport", "SciMLBase", "SciMLOperators", "SimpleNonlinearSolve", "SimpleUnPack", "SparseArrays", "SparseDiffTools", "StaticArrayInterface", "StaticArrays", "TruncatedStacktraces"]
-git-tree-sha1 = "ed171bfea6156d6458007b19790a22f4754bd501"
+git-tree-sha1 = "d6ec73f3066cf4269f762f07cec2af93c8ef4798"
uuid = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
-version = "6.71.0"
+version = "6.72.0"
[[deps.PCRE2_jll]]
deps = ["Artifacts", "Libdl"]
@@ -1742,9 +1714,9 @@ uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
[[deps.Random123]]
deps = ["Random", "RandomNumbers"]
-git-tree-sha1 = "c860e84651f58ce240dd79e5d9e055d55234c35a"
+git-tree-sha1 = "4743b43e5a9c4a2ede372de7061eed81795b12e7"
uuid = "74087812-796a-5b5d-8853-05524746bad3"
-version = "1.6.2"
+version = "1.7.0"
[[deps.RandomNumbers]]
deps = ["Random", "Requires"]
@@ -1772,9 +1744,9 @@ version = "0.6.12"
[[deps.RecursiveArrayTools]]
deps = ["Adapt", "ArrayInterface", "DocStringExtensions", "GPUArraysCore", "IteratorInterfaceExtensions", "LinearAlgebra", "RecipesBase", "SparseArrays", "StaticArraysCore", "Statistics", "SymbolicIndexingInterface", "Tables"]
-git-tree-sha1 = "09c906ce9fa905d40e0706cdb62422422091c22f"
+git-tree-sha1 = "1bbc4bb050165cc57ca2876cd53cc23395948650"
uuid = "731186ca-8d62-57ce-b412-fbd966d074cd"
-version = "3.8.1"
+version = "3.10.0"
[deps.RecursiveArrayTools.extensions]
RecursiveArrayToolsFastBroadcastExt = "FastBroadcast"
@@ -1870,9 +1842,9 @@ version = "0.6.42"
[[deps.SciMLBase]]
deps = ["ADTypes", "ArrayInterface", "CommonSolve", "ConstructionBase", "Distributed", "DocStringExtensions", "EnumX", "FillArrays", "FunctionWrappersWrappers", "IteratorInterfaceExtensions", "LinearAlgebra", "Logging", "Markdown", "PrecompileTools", "Preferences", "Printf", "RecipesBase", "RecursiveArrayTools", "Reexport", "RuntimeGeneratedFunctions", "SciMLOperators", "StaticArraysCore", "Statistics", "SymbolicIndexingInterface", "Tables", "TruncatedStacktraces"]
-git-tree-sha1 = "33e40003f4ef424e8a8700e0a3a189c6ece2af27"
+git-tree-sha1 = "375256db2d99fc730d2d134cca17939324d284d1"
uuid = "0bca4576-84f4-4d90-8ffe-ffa030f20462"
-version = "2.26.1"
+version = "2.28.0"
[deps.SciMLBase.extensions]
SciMLBaseChainRulesCoreExt = "ChainRulesCore"
@@ -1901,9 +1873,9 @@ version = "0.3.7"
[[deps.SciMLSensitivity]]
deps = ["ADTypes", "Adapt", "ArrayInterface", "ChainRulesCore", "DiffEqBase", "DiffEqCallbacks", "DiffEqNoiseProcess", "Distributions", "EllipsisNotation", "Enzyme", "FiniteDiff", "ForwardDiff", "FunctionProperties", "FunctionWrappersWrappers", "Functors", "GPUArraysCore", "LinearAlgebra", "LinearSolve", "Markdown", "OrdinaryDiffEq", "Parameters", "PreallocationTools", "QuadGK", "Random", "RandomNumbers", "RecursiveArrayTools", "Reexport", "ReverseDiff", "SciMLBase", "SciMLOperators", "SparseDiffTools", "StaticArrays", "StaticArraysCore", "Statistics", "StochasticDiffEq", "Tracker", "TruncatedStacktraces", "Zygote"]
-git-tree-sha1 = "9f536ead920f79bd0ce2ac60839c48e4a7f6eb19"
+git-tree-sha1 = "458d859240b6176c4610e79d5b75d40afa141f00"
uuid = "1ed8b502-d754-442c-8d5d-10ac956f44a1"
-version = "7.55.0"
+version = "7.56.0"
[[deps.Scratch]]
deps = ["Dates"]
@@ -2042,9 +2014,9 @@ weakdeps = ["OffsetArrays", "StaticArrays"]
[[deps.StaticArrays]]
deps = ["LinearAlgebra", "PrecompileTools", "Random", "StaticArraysCore"]
-git-tree-sha1 = "7b0e9c14c624e435076d19aea1e5cbdec2b9ca37"
+git-tree-sha1 = "bf074c045d3d5ffd956fa0a461da38a44685d6b2"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
-version = "1.9.2"
+version = "1.9.3"
weakdeps = ["ChainRulesCore", "Statistics"]
[deps.StaticArrays.extensions]
@@ -2095,9 +2067,9 @@ version = "2.0.1"
[[deps.StochasticDiffEq]]
deps = ["Adapt", "ArrayInterface", "DataStructures", "DiffEqBase", "DiffEqNoiseProcess", "DocStringExtensions", "FiniteDiff", "ForwardDiff", "JumpProcesses", "LevyArea", "LinearAlgebra", "Logging", "MuladdMacro", "NLsolve", "OrdinaryDiffEq", "Random", "RandomNumbers", "RecursiveArrayTools", "Reexport", "SciMLBase", "SciMLOperators", "SparseArrays", "SparseDiffTools", "StaticArrays", "UnPack"]
-git-tree-sha1 = "f5eb6f4794a2a56d6b9d3dcdb9d6cb217a2ac660"
+git-tree-sha1 = "97e5d0b7e5ec2e68eec6626af97c59e9f6b6c3d0"
uuid = "789caeaf-c7a9-5a7d-9973-96adeb23e2a0"
-version = "6.65.0"
+version = "6.65.1"
[[deps.StrideArraysCore]]
deps = ["ArrayInterface", "CloseOpenIntervals", "IfElse", "LayoutPointers", "ManualMemory", "SIMDTypes", "Static", "StaticArrayInterface", "ThreadingUtilities"]
@@ -2106,10 +2078,17 @@ uuid = "7792a7ef-975c-4747-a70f-980b88e8d1da"
version = "0.5.2"
[[deps.StructArrays]]
-deps = ["Adapt", "ConstructionBase", "DataAPI", "GPUArraysCore", "StaticArraysCore", "Tables"]
-git-tree-sha1 = "1b0b1205a56dc288b71b1961d48e351520702e24"
+deps = ["ConstructionBase", "DataAPI", "Tables"]
+git-tree-sha1 = "f4dc295e983502292c4c3f951dbb4e985e35b3be"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
-version = "0.6.17"
+version = "0.6.18"
+weakdeps = ["Adapt", "GPUArraysCore", "SparseArrays", "StaticArrays"]
+
+ [deps.StructArrays.extensions]
+ StructArraysAdaptExt = "Adapt"
+ StructArraysGPUArraysCoreExt = "GPUArraysCore"
+ StructArraysSparseArraysExt = "SparseArrays"
+ StructArraysStaticArraysExt = "StaticArrays"
[[deps.StructIO]]
deps = ["Test"]
@@ -2139,9 +2118,9 @@ uuid = "fb77eaff-e24c-56d4-86b1-d163f2edb164"
version = "5.2.2+0"
[[deps.SymbolicIndexingInterface]]
-git-tree-sha1 = "dc7186d456f9ff2bef0cb754a59758920f0b2382"
+git-tree-sha1 = "251bb311585143931a306175c3b7ced220300578"
uuid = "2efcf032-c050-4f8e-a9bb-153293bab1f5"
-version = "0.3.6"
+version = "0.3.8"
[[deps.SymbolicUtils]]
deps = ["AbstractTrees", "Bijections", "ChainRulesCore", "Combinatorics", "ConstructionBase", "DataStructures", "DocStringExtensions", "DynamicPolynomials", "IfElse", "LabelledArrays", "LinearAlgebra", "MultivariatePolynomials", "NaNMath", "Setfield", "SparseArrays", "SpecialFunctions", "StaticArrays", "SymbolicIndexingInterface", "TimerOutputs", "Unityper"]
@@ -2151,9 +2130,9 @@ version = "1.5.0"
[[deps.Symbolics]]
deps = ["ArrayInterface", "Bijections", "ConstructionBase", "DataStructures", "DiffRules", "Distributions", "DocStringExtensions", "DomainSets", "DynamicPolynomials", "ForwardDiff", "IfElse", "LaTeXStrings", "LambertW", "Latexify", "Libdl", "LinearAlgebra", "LogExpFunctions", "MacroTools", "Markdown", "NaNMath", "PrecompileTools", "RecipesBase", "Reexport", "Requires", "RuntimeGeneratedFunctions", "SciMLBase", "Setfield", "SparseArrays", "SpecialFunctions", "StaticArrays", "SymbolicIndexingInterface", "SymbolicUtils"]
-git-tree-sha1 = "a5d599a4a8a0671cf9944d8d43520670990f437a"
+git-tree-sha1 = "3e1e42a51142f0ee982bca641b1827cf73c84c89"
uuid = "0c5d862f-8b57-4792-8d23-62f2024744c7"
-version = "5.18.0"
+version = "5.22.0"
[deps.Symbolics.extensions]
SymbolicsGroebnerExt = "Groebner"
@@ -2353,9 +2332,9 @@ version = "1.31.0+0"
[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
-git-tree-sha1 = "801cbe47eae69adc50f36c3caec4758d2650741b"
+git-tree-sha1 = "07e470dabc5a6a4254ffebc29a1b3fc01464e105"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
-version = "2.12.2+0"
+version = "2.12.5+0"
[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
@@ -2365,9 +2344,9 @@ version = "1.1.34+0"
[[deps.XZ_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
-git-tree-sha1 = "ac88fb95ae6447c8dda6a5503f3bafd496ae8632"
+git-tree-sha1 = "37195dcb94a5970397ad425b95a9a26d0befce3a"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
-version = "5.4.6+0"
+version = "5.6.0+0"
[[deps.Xorg_libICE_jll]]
deps = ["Libdl", "Pkg"]
diff --git a/previews/PR249/assets/Project.toml b/previews/PR249/assets/Project.toml
index 39769f1..0733e33 100644
--- a/previews/PR249/assets/Project.toml
+++ b/previews/PR249/assets/Project.toml
@@ -30,7 +30,7 @@ NLopt = "0.6, 1"
Optim = "1"
Optimization = "3"
OptimizationBBO = "0.1, 0.2"
-OptimizationMOI = "0.1, 0.2, 0.3"
+OptimizationMOI = "0.1, 0.2, 0.3, 0.4"
OptimizationNLopt = "0.1, 0.2"
OptimizationOptimJL = "0.1, 0.2"
Plots = "1"
diff --git a/previews/PR249/getting_started/5b809c38.svg b/previews/PR249/getting_started/20472bab.svg
similarity index 94%
rename from previews/PR249/getting_started/5b809c38.svg
rename to previews/PR249/getting_started/20472bab.svg
index 890f47e..726774b 100644
--- a/previews/PR249/getting_started/5b809c38.svg
+++ b/previews/PR249/getting_started/20472bab.svg
@@ -1,58 +1,58 @@
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diff --git a/previews/PR249/getting_started/5eb57303.svg b/previews/PR249/getting_started/5eb57303.svg
deleted file mode 100644
index c49de67..0000000
--- a/previews/PR249/getting_started/5eb57303.svg
+++ /dev/null
@@ -1,40 +0,0 @@
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diff --git a/previews/PR249/getting_started/b0b154be.svg b/previews/PR249/getting_started/b0b154be.svg
deleted file mode 100644
index b6e0631..0000000
--- a/previews/PR249/getting_started/b0b154be.svg
+++ /dev/null
@@ -1,58 +0,0 @@
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diff --git a/previews/PR249/getting_started/c3dfece5.svg b/previews/PR249/getting_started/c3dfece5.svg
new file mode 100644
index 0000000..39d134d
--- /dev/null
+++ b/previews/PR249/getting_started/c3dfece5.svg
@@ -0,0 +1,58 @@
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diff --git a/previews/PR249/getting_started/index.html b/previews/PR249/getting_started/index.html
index 775f5a3..3b68d36 100644
--- a/previews/PR249/getting_started/index.html
+++ b/previews/PR249/getting_started/index.html
@@ -19,27 +19,27 @@
using RecursiveArrayTools # for VectorOfArray
randomized = VectorOfArray([(sol(t[i]) + 0.01randn(2)) for i in 1:length(t)])
data = convert(Array, randomized)2×200 Matrix{Float64}:
- 0.983837 1.02189 1.06586 1.10061 … 0.967578 1.00051 1.03221
- 1.00859 0.902989 0.823045 0.757726 1.10479 1.00488 0.91817Here, we used VectorOfArray from RecursiveArrayTools.jl to turn the result of an ODE into a matrix.
If we plot the solution with the parameter at a=1.42, we get the following:
newprob = remake(prob, p = [1.42])
+ 1.0227 1.02318 1.08149 1.09258 … 0.976921 1.01103 1.03312
+ 1.0062 0.90209 0.817249 0.738111 1.0885 1.00231 0.916369Here, we used VectorOfArray from RecursiveArrayTools.jl to turn the result of an ODE into a matrix.
If we plot the solution with the parameter at a=1.42, we get the following:
newprob = remake(prob, p = [1.42])
newsol = solve(newprob, Tsit5())
plot(sol)
-plot!(newsol)Notice that after one period, this solution begins to drift very far off: this problem is sensitive to the choice of a.
To build the objective function for Optim.jl, we simply call the build_loss_objective function:
cost_function = build_loss_objective(prob, Tsit5(), L2Loss(t, data),
+plot!(newsol)Notice that after one period, this solution begins to drift very far off: this problem is sensitive to the choice of a.
To build the objective function for Optim.jl, we simply call the build_loss_objective function:
cost_function = build_loss_objective(prob, Tsit5(), L2Loss(t, data),
Optimization.AutoForwardDiff(),
- maxiters = 10000, verbose = false)(::SciMLBase.OptimizationFunction{true, ADTypes.AutoForwardDiff{nothing, Nothing}, DiffEqParamEstim.var"#29#30"{Nothing, typeof(DiffEqParamEstim.STANDARD_PROB_GENERATOR), Base.Pairs{Symbol, Integer, Tuple{Symbol, Symbol}, @NamedTuple{maxiters::Int64, verbose::Bool}}, SciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, SciMLBase.ODEFunction{true, SciMLBase.AutoSpecialize, typeof(Main.f), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, L2Loss{Vector{Float64}, Matrix{Float64}, Nothing, Nothing, Nothing}, Nothing, Tuple{}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}) (generic function with 1 method)This objective function internally is calling the ODE solver to get solutions to test against the data. The keyword arguments are passed directly to the solver. Note that we set maxiters in a way that causes the differential equation solvers to error more quickly when in bad regions of the parameter space, speeding up the process. If the integrator stops early (due to divergence), then those parameters are given an infinite loss, and thus this is a quick way to avoid bad parameters. We set verbose=false because this divergence can get noisy. The Optimization.AutoForwardDiff() is a choice of automatic differentiation, i.e., how the gradients are calculated. For more information on this choice, see the automatic differentiation choice API .
A good rule of thumb is to use Optimization.AutoForwardDiff() for less than 100 parameters + states, and Optimization.AutoZygote() for more.
Before optimizing, let's visualize our cost function by plotting it for a range of parameter values:
vals = 0.0:0.1:10.0
+ maxiters = 10000, verbose = false)(::SciMLBase.OptimizationFunction{true, ADTypes.AutoForwardDiff{nothing, Nothing}, DiffEqParamEstim.var"#29#30"{Nothing, typeof(DiffEqParamEstim.STANDARD_PROB_GENERATOR), Base.Pairs{Symbol, Integer, Tuple{Symbol, Symbol}, @NamedTuple{maxiters::Int64, verbose::Bool}}, SciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, SciMLBase.ODEFunction{true, SciMLBase.AutoSpecialize, typeof(Main.f), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, SymbolicIndexingInterface.SymbolCache{Nothing, Nothing, Nothing}, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, L2Loss{Vector{Float64}, Matrix{Float64}, Nothing, Nothing, Nothing}, Nothing, Tuple{}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}) (generic function with 1 method)This objective function internally is calling the ODE solver to get solutions to test against the data. The keyword arguments are passed directly to the solver. Note that we set maxiters in a way that causes the differential equation solvers to error more quickly when in bad regions of the parameter space, speeding up the process. If the integrator stops early (due to divergence), then those parameters are given an infinite loss, and thus this is a quick way to avoid bad parameters. We set verbose=false because this divergence can get noisy. The Optimization.AutoForwardDiff() is a choice of automatic differentiation, i.e., how the gradients are calculated. For more information on this choice, see the automatic differentiation choice API .
A good rule of thumb is to use Optimization.AutoForwardDiff() for less than 100 parameters + states, and Optimization.AutoZygote() for more.
Before optimizing, let's visualize our cost function by plotting it for a range of parameter values:
vals = 0.0:0.1:10.0
plot(vals, [cost_function(i) for i in vals], yscale = :log10,
xaxis = "Parameter", yaxis = "Cost", title = "1-Parameter Cost Function",
- lw = 3)Here we see that there is a very well-defined minimum in our cost function at the real parameter (because this is where the solution almost exactly fits the dataset).
Now we can use the BFGS algorithm to optimize the parameter starting at a=1.42. We do this by creating an optimization problem and solving that with BFGS():
optprob = Optimization.OptimizationProblem(cost_function, [1.42])
-optsol = solve(optprob, BFGS())retcode: Failure
+ lw = 3)Here we see that there is a very well-defined minimum in our cost function at the real parameter (because this is where the solution almost exactly fits the dataset).
Now we can use the BFGS algorithm to optimize the parameter starting at a=1.42. We do this by creating an optimization problem and solving that with BFGS():
optprob = Optimization.OptimizationProblem(cost_function, [1.42])
+optsol = solve(optprob, BFGS())retcode: Success
u: 1-element Vector{Float64}:
- 1.5003735694001976Now let's see how well the fit performed:
newprob = remake(prob, p = optsol.u)
+ 1.500330845435876Now let's see how well the fit performed:
newprob = remake(prob, p = optsol.u)
newsol = solve(newprob, Tsit5())
plot(sol)
-plot!(newsol)Note that some algorithms may be sensitive to the initial condition. For more details on using Optim.jl, see the documentation for Optim.jl .
We can improve our solution by noting that the Lotka-Volterra equation requires that the parameters are positive. Thus, following the Optimization.jl documentation we can add box constraints to ensure the optimizer only checks between 0.0 and 3.0 which improves the efficiency of our algorithm. We pass the lb and ub keyword arguments to the OptimizationProblem to pass these bounds to the optimizer:
lower = [0.0]
+plot!(newsol)Note that some algorithms may be sensitive to the initial condition. For more details on using Optim.jl, see the documentation for Optim.jl .
We can improve our solution by noting that the Lotka-Volterra equation requires that the parameters are positive. Thus, following the Optimization.jl documentation we can add box constraints to ensure the optimizer only checks between 0.0 and 3.0 which improves the efficiency of our algorithm. We pass the lb and ub keyword arguments to the OptimizationProblem to pass these bounds to the optimizer:
lower = [0.0]
upper = [3.0]
optprob = Optimization.OptimizationProblem(cost_function, [1.42], lb = lower, ub = upper)
result = solve(optprob, BFGS())retcode: Success
u: 1-element Vector{Float64}:
- 1.5003744811807647Lastly, we can use the same tools to estimate multiple parameters simultaneously. Let's use the Lotka-Volterra equation with all parameters free:
function f2(du, u, p, t)
+ 1.500334849730979Lastly, we can use the same tools to estimate multiple parameters simultaneously. Let's use the Lotka-Volterra equation with all parameters free:
function f2(du, u, p, t)
du[1] = dx = p[1] * u[1] - p[2] * u[1] * u[2]
du[2] = dy = -p[3] * u[2] + p[4] * u[1] * u[2]
end
@@ -57,10 +57,10 @@
optprob = Optimization.OptimizationProblem(cost_function, [1.3, 0.8, 2.8, 1.2])
result_bfgs = solve(optprob, BFGS())retcode: Success
u: 4-element Vector{Float64}:
- 1.4997554523151548
- 1.0004227408198396
- 3.0003859357915856
- 1.0008823827103905The build_loss_objective with L2Loss is the most naive approach for parameter estimation. There are many others.
We can also use First-Differences in L2Loss by passing the kwarg differ_weight which decides the contribution of the differencing loss to the total loss.
cost_function = build_loss_objective(prob, Tsit5(),
+ 1.5010931152675056
+ 1.0015618867593885
+ 2.997036696996743
+ 0.9992720137993332The build_loss_objective with L2Loss is the most naive approach for parameter estimation. There are many others.
We can also use First-Differences in L2Loss by passing the kwarg differ_weight which decides the contribution of the differencing loss to the total loss.
cost_function = build_loss_objective(prob, Tsit5(),
L2Loss(t, data, differ_weight = 0.3,
data_weight = 0.7),
Optimization.AutoForwardDiff(),
@@ -68,10 +68,10 @@
optprob = Optimization.OptimizationProblem(cost_function, [1.3, 0.8, 2.8, 1.2])
result_bfgs = solve(optprob, BFGS())retcode: Failure
u: 4-element Vector{Float64}:
- 1.4995624719386118
- 1.000126610184397
- 3.0017660764437344
- 1.00102596293136We can also use Multiple Shooting method by creating a multiple_shooting_objective
function ms_f1(du, u, p, t)
+ 1.500794894914081
+ 1.0009064979879116
+ 2.9979187681753614
+ 0.9997012155328623We can also use Multiple Shooting method by creating a multiple_shooting_objective
function ms_f1(du, u, p, t)
du[1] = p[1] * u[1] - p[2] * u[1] * u[2]
du[2] = -3.0 * u[2] + u[1] * u[2]
end
@@ -89,30 +89,30 @@
ms_obj = multiple_shooting_objective(ms_prob, Tsit5(), L2Loss(t, data),
Optimization.AutoForwardDiff();
discontinuity_weight = 1.0, abstol = 1e-12,
- reltol = 1e-12)(::SciMLBase.OptimizationFunction{true, ADTypes.AutoForwardDiff{nothing, Nothing}, DiffEqParamEstim.var"#43#48"{Nothing, Float64, DiffEqParamEstim.var"#1#2", Base.Pairs{Symbol, Float64, Tuple{Symbol, Symbol}, @NamedTuple{abstol::Float64, reltol::Float64}}, SciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, SciMLBase.ODEFunction{true, SciMLBase.AutoSpecialize, typeof(Main.ms_f1), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, L2Loss{Vector{Float64}, Matrix{Float64}, Nothing, Nothing, Nothing}, Nothing}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}) (generic function with 1 method)This creates the objective function that can be passed to an optimizer, from which we can then get the parameter values and the initial values of the short time periods, keeping in mind the indexing. Now we mix this with a global optimization method to improve robustness even more:
optprob = Optimization.OptimizationProblem(ms_obj, zeros(18), lb = first.(bound),
+ reltol = 1e-12)(::SciMLBase.OptimizationFunction{true, ADTypes.AutoForwardDiff{nothing, Nothing}, DiffEqParamEstim.var"#43#48"{Nothing, Float64, DiffEqParamEstim.var"#1#2", Base.Pairs{Symbol, Float64, Tuple{Symbol, Symbol}, @NamedTuple{abstol::Float64, reltol::Float64}}, SciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, SciMLBase.ODEFunction{true, SciMLBase.AutoSpecialize, typeof(Main.ms_f1), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, SymbolicIndexingInterface.SymbolCache{Nothing, Nothing, Nothing}, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, L2Loss{Vector{Float64}, Matrix{Float64}, Nothing, Nothing, Nothing}, Nothing}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}) (generic function with 1 method)This creates the objective function that can be passed to an optimizer, from which we can then get the parameter values and the initial values of the short time periods, keeping in mind the indexing. Now we mix this with a global optimization method to improve robustness even more:
optprob = Optimization.OptimizationProblem(ms_obj, zeros(18), lb = first.(bound),
ub = last.(bound))
optsol_ms = solve(optprob, BBO_adaptive_de_rand_1_bin_radiuslimited(), maxiters = 10_000)retcode: Failure
u: 18-element Vector{Float64}:
- 0.911322656937296
- 1.1266787773646716
- 3.5006585319727703
- 0.2581565028106773
- 2.6729470845263212
- 4.695872235004817
- 1.3355182732930886
- 0.8057052583335481
- 4.8427140131072415
- 0.37306985699367085
- 1.2771819371346613
- 3.520691733864322
- 1.7035243914223086
- 0.4306520080126741
- 6.388765407461336
- 0.7747149171072861
- 1.498912664368806
- 1.0264052358197866optsol_ms.u[(end - 1):end]2-element Vector{Float64}:
- 1.498912664368806
- 1.0264052358197866Here as our model had 2 parameters, we look at the last 2 indexes of result to get our parameter values and the rest of the values are the initial values of the shorter timespans as described in the reference section. We can also use a gradient-based optimizer with the multiple shooting objective.
optsol_ms = solve(optprob, BFGS())
+ 0.8956492828785643
+ 1.3160211458098436
+ 3.2036079521090404
+ 0.34414204633933954
+ 2.9690542283041736
+ 4.679574111928217
+ 1.114899259968644
+ 0.7867338732898105
+ 4.7282992464471
+ 0.39043062426262265
+ 1.4498868316927416
+ 3.709004168905337
+ 1.5496759773692697
+ 0.6687725955065645
+ 6.191636920691155
+ 0.824008168705664
+ 1.7032464763211117
+ 1.0632935540792416optsol_ms.u[(end - 1):end]2-element Vector{Float64}:
+ 1.7032464763211117
+ 1.0632935540792416Here as our model had 2 parameters, we look at the last 2 indexes of result to get our parameter values and the rest of the values are the initial values of the shorter timespans as described in the reference section. We can also use a gradient-based optimizer with the multiple shooting objective.
optsol_ms = solve(optprob, BFGS())
optsol_ms.u[(end - 1):end]2-element Vector{Float64}:
1.5000000000037266
1.000000000000063The objective function for the Two Stage method can be created and passed to an optimizer as
two_stage_obj = two_stage_objective(ms_prob, t, data, Optimization.AutoForwardDiff())
@@ -122,4 +122,4 @@
1.5035938533664905
0.99257311537469
2.8
- 1.2The default kernel used in the method is Epanechnikov, available others are Uniform, Triangular, Quartic, Triweight, Tricube, Gaussian, Cosine, Logistic and Sigmoid, this can be passed by the kernel keyword argument. loss_func keyword argument can be used to pass the loss function (cost function) you want to use and passing a valid adtype argument
There are many more choices for how to improve the robustness of a parameter estimation. With all of these tools, one likely should never do the simple “solve it with p and check the L2 loss”. Instead, we should use these tricks to improve the loss landscape and increase the ability for optimizers to find globally the best parameters.
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