A neural network assisted Metropolis adjusted Langevin algorithm

Christian Müller; Holger Diedam; Thomas Mrziglod; Andreas Schuppert

doi:10.1515/mcma-2020-2060

Published by De Gruyter April 15, 2020

A neural network assisted Metropolis adjusted Langevin algorithm

Christian Müller , Holger Diedam , Thomas Mrziglod and Andreas Schuppert

From the journal Monte Carlo Methods and Applications

https://doi.org/10.1515/mcma-2020-2060

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Abstract

In this paper, we derive a Markov chain Monte Carlo (MCMC) algorithm supported by a neural network. In particular, we use the neural network to substitute derivative calculations made during a Metropolis adjusted Langevin algorithm (MALA) step with inexpensive neural network evaluations. Using a complex, high-dimensional blood coagulation model and a set of measurements, we define a likelihood function on which we evaluate the new MCMC algorithm. The blood coagulation model is a dynamic model, where derivative calculations are expensive and hence limit the efficiency of derivative-based MCMC algorithms. The MALA adaptation greatly reduces the time per iteration, while only slightly affecting the sample quality. We also test the new algorithm on a 2-dimensional example with a non-convex shape, a case where the MALA algorithm has a clear advantage over other state of the art MCMC algorithms. To assess the impact of the new algorithm, we compare the results to previously generated results of the MALA and the random walk Metropolis Hastings (RWMH).

Keywords: Markov chain Monte Carlo; neural network

MSC 2010: 65C05

A Appendix: A blood coagulation model

What follows is the mathematical formulation of the ODE system of the blood coagulation model we use in Section 4.1. For a detailed, biological description of the parameters, we refer to [6] and the added supplementary materials.

d⁢x0d⁢t=u81⋅x2+u83⋅x26-u82⋅x0-p19⋅x0⋅x3,

d⁢x1d⁢t=u81⋅x2-p20⋅x1⋅(x27+x28)-u82⋅x0⋅x1-u84⋅x1⋅x26-u85⋅x1⋅x29,

d⁢x2d⁢t=u82⋅x0⋅x1-u81⋅x2,

d⁢x3d⁢t=u83⋅x26+p20⋅x1⋅(x27+x28)-p19⋅x0⋅x3+u84⋅x1⋅x26+u85⋅x1⋅x29,

d⁢x4d⁢t=p21⋅x31+u78⋅p16⋅x30-p22⋅x4⋅x26-u78⋅x4⋅x90,

d⁢x5d⁢t=u87⋅x34+u77⋅p15⋅x33-u88⋅x5⋅x26-u77⋅x5⋅x90-p38⋅x5⋅x63p39+x5,

d⁢x6d⁢t=p31⋅x47+u73⋅p13⋅x7-u90⋅x6⋅(x27+x28)-u73⋅x6⋅x90-p32⋅x6⋅x46,

d⁢x7d⁢t=u73⋅x6⋅x90-u90⋅x7⋅(x27+x28)-u73⋅p13⋅x7-p32⋅x7⋅x46,

d⁢x8d⁢t=u204⋅x121+u76⋅p14⋅x37-x8⋅(u91⋅(x27+x28)+p25⋅x29+p25⋅x48)-u76⋅x8⋅x90-p49⋅x8⋅x120,

d⁢x9d⁢t=u74⋅u75⋅x44-x9⋅(u99⋅(x27+x28)+u98⋅x29+p28⋅x48)-u74⋅x9⋅x90,

d⁢x10d⁢t=u100⋅x49+u102⋅x50-u101⋅x10⋅x27-p35⋅x10⋅x32,

d⁢x11d⁢t=u212⋅u213⋅x83-u104⋅x11⋅x27-u107⋅x11⋅x26-u106⋅x11⋅x29-u105⋅x11⋅x35-p36⋅x11⋅x48-u110⋅x11⋅x58-u126⋅x11⋅x63-u153⋅x11⋅x84-p45⋅x11⋅x70-u179⋅x11⋅x72-u212⋅x11⋅x124-p45⋅u201⋅x11⋅x115,

d⁢x12d⁢t=-u117⋅x12⋅(x29+x48⋅0.07)u118+x12,

d⁢x13d⁢t=-u124⋅x13⋅x63,

d⁢x14d⁢t=u79⋅p17⋅x15-u79⋅x14⋅x90-p37⋅x14⋅(x29+x48)u121+x14-u119⋅x14⋅x89u120+x14,

d⁢x15d⁢t=u79⋅x14⋅x90-u79⋅p17⋅x15-p37⋅x15⋅(x29+x48)u121+x15-u119⋅x15⋅x89u120+x15,

d⁢x16d⁢t=-u125⋅x16⋅x63-u177⋅x16⋅x27-p44⋅x16⋅x29-p44⋅x16⋅x48-u178⋅x16⋅x72,

d⁢x17d⁢t=-u127⋅x17⋅x63-u165⋅x17⋅x72,

d⁢x18d⁢t=-u128⋅x18⋅x63,

d⁢x19d⁢t=u140⋅x118+x9⋅(u99⋅(x27+x28)+u98⋅x29+p28⋅x48)+u74⋅u75⋅x45-u74⋅x19⋅x90-u133⋅u209⋅x19⋅x73,

d⁢x20d⁢t=u108⋅u116⋅x58+u108⋅u116⋅x59+u109⋅u115⋅x60+u111⋅u112⋅x61+u113⋅u114⋅x78-p0⋅u114⋅x20-u108⋅x20⋅x27-u108⋅x20⋅x28-u109⋅x20⋅x46-u111⋅x20⋅x51,

d⁢x21d⁢t=u142⋅x79+u144⋅x80+u146⋅x82-p0⋅u147⋅x21+u147⋅u148⋅x81-u141⋅x21⋅x29-u143⋅x21⋅x48-u145⋅x21⋅x70,

d⁢x22d⁢t=u150⋅x84+u150⋅x85+u152⋅x86+u155⋅x88+u156⋅u157⋅x87-p0⋅u156⋅x22-u149⋅x22⋅x27-u149⋅x22⋅x28-u151⋅x22⋅x46-u154⋅x22⋅x51,

d⁢x23d⁢t=u137⋅x71+u137⋅x116-u129⋅x23⋅x70-u129⋅x23⋅x115,

d⁢x24d⁢t=u131⋅x73-u130⋅x24⋅x72,

d⁢x25d⁢t=u136⋅x70+u136⋅x115+p45⋅x11⋅x70+u172⋅x70⋅x94+p45⋅u201⋅x11⋅x115+u172⋅u201⋅x94⋅x115-p40⋅x25⋅x29-p40⋅x25⋅x48,

d⁢x26d⁢t=p21⋅x31⋅2.0+p24⋅x32⋅2.0+u87⋅x34+u89⋅x34+u87⋅x91+u89⋅x91+p19⋅x0⋅x3-u83⋅x26-p22⋅x4⋅x26-u88⋅x5⋅x26-p22⋅x26⋅x30-u86⋅x26⋅x27-u86⋅x26⋅x28-u88⋅x26⋅x33-u107⋅x11⋅x26-u103⋅x26⋅x49,

d⁢x27d⁢t=p24⋅x32+p29⋅x46+u100⋅x49+u150⋅x84+u78⋅p16⋅x28+u108⋅u116⋅x58+u166⋅u167⋅x103-u86⋅x26⋅x27-u101⋅x10⋅x27-u104⋅x11⋅x27-u108⋅x20⋅x27-p30⋅x27⋅x45-u78⋅x27⋅x90-u149⋅x22⋅x27-u177⋅x16⋅x27-u162⋅x27⋅x93-u166⋅x27⋅x94-u215⋅x27⋅x83,

d⁢x28d⁢t=p24⋅x32+u94⋅x41+u150⋅x85+u108⋅u116⋅x59+u78⋅x27⋅x90+u122⋅x30⋅x36u123+x30-u86⋅x26⋅x28-u108⋅x20⋅x28-p30⋅x28⋅x45-u78⋅p16⋅x28-u149⋅x22⋅x28-u215⋅x28⋅x83,

d⁢x29d⁢t=u136⋅x70+u142⋅x79+u90⋅x6⋅(x27+x28)+u90⋅x7⋅(x27+x28)+u168⋅u169⋅x105+u173⋅u174⋅x108-u106⋅x11⋅x29+p34⋅x46⋅x48-p40⋅x25⋅x29-u141⋅x21⋅x29-p44⋅x16⋅x29-p43⋅x29⋅x92-u163⋅x29⋅x93-u168⋅x29⋅x94-u173⋅x29⋅x95-u214⋅x29⋅x83,

d⁢x30d⁢t=p21⋅x31+u93⋅x41+u97⋅x41+u78⋅x4⋅x90-u78⋅p16⋅x30-p22⋅x26⋅x30-p27⋅x30⋅x40-u122⋅x30⋅x36u123+x30,

d⁢x31d⁢t=p21⋅x31⋅(-2.0)+p22⋅x4⋅x26+p22⋅x26⋅x30-p23⋅x31,

d⁢x32d⁢t=p23⋅x31-p24⋅x32⋅2.0+u102⋅x50+u86⋅x26⋅x27+u86⋅x26⋅x28-p35⋅x10⋅x32,

d⁢x33d⁢t=u87⋅x91+u77⋅x5⋅x90-u77⋅p15⋅x33-u88⋅x26⋅x33-p38⋅x33⋅x64p39+x33,

d⁢x34d⁢t=u88⋅x5⋅x26-u87⋅x34-u89⋅x34,

d⁢x35d⁢t=u89⋅x34+u77⋅p15⋅x36+p38⋅x5⋅x63p39+x5-u105⋅x11⋅x35-u77⋅x35⋅x90-u217⋅x35⋅x83,

d⁢x36d⁢t=u92⋅x40+u97⋅x40+u97⋅x41+u89⋅x91+u77⋅x35⋅x90+p38⋅x33⋅x64p39+x33-u77⋅p15⋅x36-p26⋅x36⋅x39-u217⋅x36⋅x83,

d⁢x37d⁢t=-x37⋅(u91⋅(x27+x28)+p25⋅x29+p25⋅x48)-u76⋅p14⋅x37+u76⋅x8⋅x90,

d⁢x38d⁢t=u139⋅x119+u206⋅x122+x8⋅(u91⋅(x27+x28)+p25⋅x29+p25⋅x48)+u76⋅p14⋅x39-u76⋅x38⋅x90-u205⋅x38⋅x120-u132⋅u209⋅x38⋅x73,

d⁢x39d⁢t=u92⋅x40+u139⋅x74+x37⋅(u91⋅(x27+x28)+p25⋅x29+p25⋅x48)+u95⋅x42⋅x43+u76⋅x38⋅x90-u96⋅x39-u76⋅p14⋅x39-p26⋅x36⋅x39-u132⋅x39⋅x73,

d⁢x40d⁢t=u93⋅x41+u94⋅x41+p26⋅x36⋅x39-u92⋅x40-u97⋅x40-p27⋅x30⋅x40,

d⁢x41d⁢t=p27⋅x30⋅x40-u93⋅x41-u94⋅x41-u97⋅x41,

d⁢x42d⁢t=u96⋅x39+u97⋅x40+u97⋅x41-u95⋅x42⋅x43,

d⁢x43d⁢t=u96⋅x39+u97⋅x40+u97⋅x41-u95⋅x42⋅x43,

d⁢x44d⁢t=u74⋅x9⋅x90-x44⋅(u99⋅(x27+x28)+u98⋅x29+p28⋅x48)-u74⋅u75⋅x44,

d⁢x45d⁢t=p29⋅x46+u140⋅x75+x44⋅(u99⋅(x27+x28)+u98⋅x29+p28⋅x48)+u74⋅x19⋅x90+u218⋅x46⋅x83-u74⋅u75⋅x45-p30⋅x27⋅x45-p30⋅x28⋅x45-u133⋅x45⋅x73,

d⁢x46d⁢t=p31⋅x47+p33⋅x47+u152⋅x86+u109⋅u115⋅x60+p30⋅x27⋅x45+p30⋅x28⋅x45-p29⋅x46-p32⋅x6⋅x46-p32⋅x7⋅x46-u109⋅x20⋅x46-u151⋅x22⋅x46-u218⋅x46⋅x83,

d⁢x47d⁢t=p32⋅x6⋅x46+p32⋅x7⋅x46-p31⋅x47-p33⋅x47,

d⁢x48d⁢t=p33⋅x47+u144⋅x80+u136⋅x115+u168⋅u169⋅x104+u173⋅u174⋅x107-p36⋅x11⋅x48-p34⋅x46⋅x48-p40⋅x25⋅x48-u143⋅x21⋅x48-p44⋅x16⋅x48-p43⋅x48⋅x92-u163⋅x48⋅x93-u168⋅x48⋅x94-u173⋅x48⋅x95-u214⋅x48⋅x83,

d⁢x49d⁢t=+u101⋅x10⋅x27-u100⋅x49-u103⋅x26⋅x49,

d⁢x50d⁢t=+p35⋅x10⋅x32+u103⋅x26⋅x49-u102⋅x50,

d⁢x51d⁢t=u155⋅x88+u111⋅u112⋅x61+u104⋅x11⋅x27+u215⋅x27⋅x83+u215⋅x28⋅x83+u218⋅x46⋅x83-u111⋅x20⋅x51-u154⋅x22⋅x51,

d⁢x52d⁢t=0,

d⁢x53d⁢t=p36⋅x11⋅x48+u214⋅x48⋅x83+p45⋅u201⋅x11⋅x115,

d⁢x54d⁢t=u105⋅x11⋅x35+u217⋅x35⋅x83+u217⋅x36⋅x83,

d⁢x55d⁢t=0,

d⁢x56d⁢t=u106⋅x11⋅x29+p45⋅x11⋅x70+u214⋅x29⋅x83,

d⁢x57d⁢t=u107⋅x11⋅x26,

d⁢x58d⁢t=u78⋅p16⋅x59+u108⋅x20⋅x27-u108⋅u116⋅x58-u110⋅x11⋅x58-u78⋅x58⋅x90,

d⁢x59d⁢t=u108⋅x20⋅x28+u78⋅x58⋅x90-u78⋅p16⋅x59-u108⋅u116⋅x59,

d⁢x60d⁢t=u109⋅x20⋅x46-u109⋅u115⋅x60,

d⁢x61d⁢t=u110⋅x11⋅x58+u111⋅x20⋅x51-u111⋅u112⋅x61,

d⁢x62d⁢t=u117⋅x12⋅(x29+x48⋅0.07)u118+x12,

d⁢x63d⁢t=u80⋅p18⋅x64+p37⋅x14⋅(x29+x48)u121+x14+u119⋅x14⋅x89u120+x14-u124⋅x13⋅x63-u126⋅x11⋅x63-u125⋅x16⋅x63-u127⋅x17⋅x63-u128⋅x18⋅x63-u80⋅x63⋅x90-u216⋅x63⋅x83,

d⁢x64d⁢t=u80⋅x63⋅x90+p37⋅x15⋅(x29+x48)u121+x15+u119⋅x15⋅x89u120+x15-u80⋅p18⋅x64-u216⋅x64⋅x83,

d⁢x65d⁢t=u124⋅x13⋅x63,

d⁢x66d⁢t=u125⋅x16⋅x63,

d⁢x67d⁢t=u126⋅x11⋅x63+u216⋅x63⋅x83+u216⋅x64⋅x83,

d⁢x68d⁢t=u127⋅x17⋅x63,

d⁢x69d⁢t=u128⋅x18⋅x63,

d⁢x70d⁢t=u137⋅x71+u138⋅x71+u146⋅x82+p40⋅x25⋅x29-u136⋅x70-u129⋅x23⋅x70-u145⋅x21⋅x70-p45⋅x11⋅x70-u172⋅x70⋅x94,

d⁢x71d⁢t=u129⋅x23⋅x70-u137⋅x71-u138⋅x71,

d⁢x72d⁢t=u131⋅x73+u138⋅x71+u170⋅u171⋅x106+u175⋅u176⋅x109+u138⋅u208⋅x116-u130⋅x24⋅x72-u165⋅x17⋅x72-u179⋅x11⋅x72-u178⋅x16⋅x72-u164⋅x72⋅x93-u170⋅x72⋅x94-u175⋅x72⋅x95,

d⁢x73d⁢t=+u134⋅x75+u135⋅x74+u139⋅x74+u140⋅x75+u134⋅x118+u135⋅x119+u139⋅x119+u140⋅x118+u130⋅x24⋅x72-u131⋅x73-u132⋅x39⋅x73-u133⋅x45⋅x73-u133⋅u209⋅x19⋅x73-u132⋅u209⋅x38⋅x73,

d⁢x74d⁢t=u132⋅x39⋅x73-u135⋅x74-u139⋅x74,

d⁢x75d⁢t=u133⋅x45⋅x73-u134⋅x75-u140⋅x75,

d⁢x76d⁢t=u134⋅x75+u134⋅x118,

d⁢x77d⁢t=u135⋅x74+u135⋅x119,

d⁢x78d⁢t=p0⋅u114⋅x20-u113⋅u114⋅x78,

d⁢x79d⁢t=u141⋅x21⋅x29-u142⋅x79,

d⁢x80d⁢t=u143⋅x21⋅x48-u144⋅x80,

d⁢x81d⁢t=p0⋅u147⋅x21-u147⋅u148⋅x81,

d⁢x82d⁢t=u145⋅x21⋅x70-u146⋅x82,

d⁢x83d⁢t=u212⋅x11⋅x124-u212⋅u213⋅x83-u215⋅x27⋅x83-u214⋅x29⋅x83-u215⋅x28⋅x83-u217⋅x35⋅x83-u217⋅x36⋅x83-u214⋅x48⋅x83-u218⋅x46⋅x83-u216⋅x63⋅x83-u216⋅x64⋅x83,

d⁢x84d⁢t=u78⋅p16⋅x85+u149⋅x22⋅x27-u150⋅x84-u78⋅x84⋅x90-u153⋅x11⋅x84,

d⁢x85d⁢t=u149⋅x22⋅x28+u78⋅x84⋅x90-u150⋅x85-u78⋅p16⋅x85,

d⁢x86d⁢t=u151⋅x22⋅x46-u152⋅x86,

d⁢x87d⁢t=p0⋅u156⋅x22-u156⋅u157⋅x87,

d⁢x88d⁢t=u154⋅x22⋅x51+u153⋅x11⋅x84-u155⋅x88,

d⁢x89d⁢t=0,

d⁢x90d⁢t=u96⋅x39+u134⋅x75+u135⋅x74+u90⋅x7⋅(x27+x28)+u73⋅p13⋅x7+u79⋅p17⋅x15+u78⋅p16⋅x28+u76⋅p14⋅x37+u77⋅p15⋅x33+u78⋅p16⋅x30+u76⋅p14⋅x39+u74⋅u75⋅x44+u77⋅p15⋅x36+u74⋅u75⋅x45+u78⋅p16⋅x59+u80⋅p18⋅x64+u78⋅p16⋅x85+p22⋅x26⋅x30+u86⋅x26⋅x28+p32⋅x7⋅x46+p27⋅x30⋅x40+p26⋅x36⋅x39+p30⋅x28⋅x45+u215⋅x28⋅x83+u217⋅x36⋅x83+u216⋅x64⋅x83-p21⋅x31-p24⋅x32-u92⋅x40-u93⋅x41-u94⋅x41-u97⋅x41-u73⋅x6⋅x90-u74⋅x9⋅x90-u77⋅x5⋅x90-u76⋅x8⋅x90-u78⋅x4⋅x90-u74⋅x19⋅x90-u79⋅x14⋅x90-u95⋅x42⋅x43-u78⋅x27⋅x90-u77⋅x35⋅x90-u76⋅x38⋅x90-u78⋅x58⋅x90-u80⋅x63⋅x90-u78⋅x84⋅x90,

d⁢x91d⁢t=u88⋅x26⋅x33-u87⋅x91-u89⋅x91,

d⁢x92d⁢t=-p43⋅x29⋅x92-p43⋅x48⋅x92,

d⁢x93d⁢t=-u162⋅x27⋅x93-u163⋅x29⋅x93-u163⋅x48⋅x93-u164⋅x72⋅x93,

d⁢x94d⁢t=u166⋅u167⋅x103+u168⋅u169⋅x104+u168⋅u169⋅x105+u170⋅u171⋅x106-u166⋅x27⋅x94-u168⋅x29⋅x94-u168⋅x48⋅x94-u170⋅x72⋅x94-u172⋅x70⋅x94-u172⋅u201⋅x94⋅x115,

d⁢x95d⁢t=u173⋅u174⋅x107+u173⋅u174⋅x108+u175⋅u176⋅x109-u173⋅x29⋅x95-u173⋅x48⋅x95-u175⋅x72⋅x95,

d⁢x96d⁢t=u179⋅x11⋅x72,

d⁢x97d⁢t=p43⋅x48⋅x92,

d⁢x98d⁢t=p43⋅x29⋅x92,

d⁢x99d⁢t=u162⋅x27⋅x93,

d⁢x100d⁢t=u163⋅x48⋅x93,

d⁢x101d⁢t=u163⋅x29⋅x93,

d⁢x102d⁢t=u164⋅x72⋅x93,

d⁢x103d⁢t=u166⋅x27⋅x94-u166⋅u167⋅x103,

d⁢x104d⁢t=u168⋅x48⋅x94+u172⋅u201⋅x94⋅x115-u168⋅u169⋅x104,

d⁢x105d⁢t=u168⋅x29⋅x94+u172⋅x70⋅x94-u168⋅u169⋅x105,

d⁢x106d⁢t=u170⋅x72⋅x94-u170⋅u171⋅x106,

d⁢x107d⁢t=u173⋅x48⋅x95-u173⋅u174⋅x107,

d⁢x108d⁢t=u173⋅x29⋅x95-u173⋅u174⋅x108,

d⁢x109d⁢t=u175⋅x72⋅x95-u175⋅u176⋅x109,

d⁢x110d⁢t=u177⋅x16⋅x27,

d⁢x111d⁢t=p44⋅x16⋅x48,

d⁢x112d⁢t=p44⋅x16⋅x29,

d⁢x113d⁢t=u178⋅x16⋅x72,

d⁢x114d⁢t=u165⋅x17⋅x72,

d⁢x115d⁢t=+u137⋅x116+u138⋅u208⋅x116+p40⋅x25⋅x48-u136⋅x115-u129⋅x23⋅x115-p45⋅u201⋅x11⋅x115-u172⋅u201⋅x94⋅x115,

d⁢x116d⁢t=u129⋅x23⋅x115-u137⋅x116-u138⋅u208⋅x116,

d⁢x117d⁢t=0,

d⁢x118d⁢t=u133⋅u209⋅x19⋅x73-u134⋅x118-u140⋅x118,

d⁢x119d⁢t=u132⋅u209⋅x38⋅x73-u135⋅x119-u139⋅x119,

d⁢x120d⁢t=u204⋅x121+u206⋅x122-p49⋅x8⋅x120-u205⋅x38⋅x120,

d⁢x121d⁢t=p49⋅x8⋅x120-u204⋅x121-u207⋅x121⋅(x29+x48),

d⁢x122d⁢t=u207⋅x121⋅(x29+x48)+u205⋅x38⋅x120-u206⋅x122,

d⁢x123d⁢t=p0⋅u219⋅x124-u219⋅u220⋅x123,

d⁢x124d⁢t=+u212⋅u213⋅x83+u219⋅u220⋅x123+u215⋅x27⋅x83+u214⋅x29⋅x83+u215⋅x28⋅x83+u217⋅x35⋅x83+u217⋅x36⋅x83+u214⋅x48⋅x83+u218⋅x46⋅x83+u216⋅x63⋅x83+u216⋅x64⋅x83-p0⋅u219⋅x124-u212⋅x11⋅x124.

The initial values of the system are given as IV dependent on the constant parameters u and variable parameters p.

IV0=p0⋅p1, ⁢IV1=p0⋅p2⋅p42, ⁢IV2=p0⋅u0,

IV3=p0⋅p3⋅p42, ⁢IV4=p0⋅p4⋅p42, ⁢IV5=p0⋅p5⋅p42,

IV6=p0⋅p6⋅p42, ⁢IV7=p0⋅u1, ⁢IV8=p0⋅p7,

IV9=p0⋅p8, ⁢IV10=p0⋅p9, ⁢IV11=p0⋅p10,

IV12=p0⋅u2, ⁢IV13=p0⋅u3, ⁢IV14=p0⋅u4,

IV15=p0⋅u5, ⁢IV16=p0⋅p11, ⁢IV17=p0⋅u6,

IV18=p0⋅u7, ⁢IV19=p0⋅(u8+p8⋅p41), ⁢IV20=p0⋅u113⋅u158u133+1.0,

IV21=p0⋅u148⋅u159u148+1.0, ⁢IV22=p0⋅u157⋅u160u157+1.0, ⁢IV23=p0⋅u9⋅p42,

IV24=p0⋅u10⋅p42, ⁢IV25=p0⋅p12, ⁢IV26=p0⋅u11,

IV27=p0⋅u12⋅p42, ⁢IV28=p0⋅u13, ⁢IV29=p0⋅u14⋅p42,

IV30=p0⋅u15, ⁢IV31=p0⋅u16, ⁢IV32=p0⋅u17,

IV33=p0⋅u18, ⁢IV34=p0⋅u19, ⁢IV35=p0⋅u20⋅p42,

IV36=p0⋅u21, ⁢IV37=p0⋅u22, ⁢IV38=p0⋅u23,

IV39=p0⋅u24, ⁢IV40=p0⋅u25, ⁢IV41=p0⋅u26,

IV42=p0⋅u27, ⁢IV43=p0⋅u28, ⁢IV44=p0⋅u29,

IV45=p0⋅u30, ⁢IV46=p0⋅u31, ⁢IV47=p0⋅u32,

IV48=p0⋅u33, ⁢IV49=p0⋅u34, ⁢IV50=p0⋅u35,

IV51=p0⋅u36, ⁢IV52=p0⋅u37, ⁢IV53=p0⋅u38,

IV54=p0⋅u39, ⁢IV55=p0⋅u40, ⁢IV56=p0⋅u41,

IV57=p0⋅u42, ⁢IV58=p0⋅u43, ⁢IV59=p0⋅u44,

IV60=p0⋅u45, ⁢IV61=p0⋅u46, ⁢IV62=p0⋅u47,

IV63=p0⋅u48, ⁢IV64=p0⋅u49, ⁢IV65=p0⋅u50,

IV66=p0⋅u51, ⁢IV67=p0⋅u52, ⁢IV68=p0⋅u53,

IV69=p0⋅u54, ⁢IV70=p0⋅u55, ⁢IV71=p0⋅u56,

IV72=p0⋅u57, ⁢IV73=p0⋅u58, ⁢IV74=p0⋅u59,

IV75=p0⋅u60, ⁢IV76=p0⋅u61, ⁢IV77=p0⋅u62,

IV78=p0⋅u158u113+1.0, ⁢IV79=p0⋅u63, ⁢IV80=p0⋅u64,

IV81=p0⋅u8u148+1.0, ⁢IV82=p0⋅u65, ⁢IV83=p0⋅u66,

IV84=p0⋅u67, ⁢IV85=p0⋅u68⋅p42, ⁢IV86=p0⋅u69⋅p42,

IV87=p0⋅u160u157+1.0, ⁢IV88=p0⋅u70, ⁢IV89=p0⋅u71⋅p42,

IV90=p50, ⁢IV91=p0⋅u72⋅p42, ⁢IV92=p0⋅p46,

IV93=p0⋅u180, ⁢IV94=p0⋅p47, ⁢IV95=p0⋅u181,

IV96=p0⋅u182, ⁢IV97=p0⋅u183, ⁢IV98=p0⋅u184,

IV99=p0⋅u185, ⁢IV100=p0⋅u186, ⁢IV101=p0⋅u187,

IV102=p0⋅u188, ⁢IV103=p0⋅u189, ⁢IV104=p0⋅u190,

IV105=p0⋅u191, ⁢IV106=p0⋅u192, ⁢IV107=p0⋅u193,

IV108=p0⋅u194, ⁢IV109=p0⋅u195, ⁢IV110=p0⋅u196,

IV111=p0⋅u197, ⁢IV112=p0⋅u198, ⁢IV113=p0⋅u199,

IV114=p0⋅u200, ⁢IV115=p0⋅u223, ⁢IV116=p0⋅u224,

IV117=0, ⁢IV118=p0⋅u221, ⁢IV119=p0⋅u222,

IV120=p0⋅p48, ⁢IV121=p0⋅u202, ⁢IV122=p0⋅u203,

IV123=p0⋅u211⋅(u161+u210)u220+1.0, ⁢IV124=p0⋅u211⋅u220⋅(u161+u210)u220+1.0.

Table 3

List of fixed parameters (1/2).

p0	Dilutionfactor	0.6667	p1	S_TF	7.658989e-12	p2	S_VII	1.561404e-08
p3	S_VIIa	5.326683e-11	p4	S_X	2.039697e-07	p5	S_IX	9.096479e-08
p6	S_II	2.017022e-06	p7	S_VIII	7.518441e-10	p8	S_V	1.886433e-08
p9	S_TFPI	2.307784e-09	p10	S_ATIII	2.661478e-06	p11	S_alpha1AT	2.37676e-05
p12	S_Tm	9.79467e-09	p13	Kd2	-15.04544	p14	Kd8	3.489801e-09
p15	Kd9	2.426951e-09	p16	Kd10	2.535596e-07	p17	Kd11	-18.43114
p18	Kd11a	1.602839e-09	p19	k4	26364809	p20	k6	16759672
p21	k8	1.578925	p22	k9	32443880	p23	k10	1.828158
p24	k11	2.486723	p25	k17	15.61378	p26	k19	17.88283
p27	k21	811283301	p28	k26b	15.5222	p29	k27	0.3163996
p30	k28	21.5559	p31	k29	136.4629	p32	k30	30080707
p33	k31	104.459	p34	k32	53068894	p35	k36	240193650
p36	k39	88172.98	p37	Kog10	-3.772776	p38	Kog8	0.2416168
p39	Kog8m	4.0979e-07	p40	kbu50	14.02588	p41	Va_pre_ActofV	-4.503533
p42	WarfarinFactor	0.5978258	p43	Veer1	1537.89	p44	Down2	89.12773
p45	Bourin1	10.3116	p46	S_HCII	1.646771e-06	p47	S_PCI	8.728542e-08
p48	S_vWF	3.707884e-08	p49	kvWF1	583672.2	p50	S_PL	2.282184e-08

Table 4

List of fixed parameters (2/2).

u0	0	u1	0	u2	7.091038e-06	u3	1.201881e-06	u4	3.287149e-08
u5	0	u6	9.586261e-07	u7	4.885827e-10	u8	0	u9	5.867748e-08
u10	5.268007e-08	u11	0	u12	1.381180e-14	u13	0	u14	0
u15	0	u16	0	u17	0	u18	0	u19	0
u20	0	u21	0	u22	0	u23	0	u24	0
u25	0	u26	0	u27	0	u28	0	u29	0
u30	0	u31	0	u32	0	u33	0	u34	0
u35	0	u36	0	u37	0	u38	0	u39	0
u40	0	u41	0	u42	0	u43	0	u44	0
u45	0	u46	0	u47	0	u48	0	u49	0
u50	0	u51	0	u52	0	u53	0	u54	0
u55	0	u56	0	u57	0	u58	0	u59	0
u60	0	u61	0	u62	0	u63	0	u64	0
u65	0	u66	0	u67	0	u68	0	u69	0
u70	0	u71	0	u72	0	u73	1.233638e+07	u74	1.932636e+06
u75	6.253566e-10	u76	5.787158e+07	u77	5.551706e+07	u78	1.350195e+06	u79	6.886843e+06
u80	4.219812e+06	u81	3.486538e-03	u82	9.838915e+04	u83	3.343605e-03	u84	4.907052e+05
u85	1.659267e+04	u86	2.034925e+07	u87	2.463716	u88	1.453219e+07	u89	3.667853e-01
u90	6.113555e+04	u91	7.604036e+05	u92	5.642855e-03	u93	5.850895e-04	u94	9.693666e+01
u95	2.333402e+04	u96	6.103863e-03	u97	4.875274e-04	u98	2.475065e+06	u99	5.122565e+04
u100	3.969542e-04	u101	8.620074e+05	u102	7.423331e-03	u103	4.225866e+07	u104	2.037410e+03
u105	4.950986e+02	u106	3.683459e+02	u107	2.290561e+02	u108	6.109323e+07	u109	1.717579e+06
u110	0	u111	0	u112	3.370790e-10	u113	5.122001e-02	u114	1.745986e-01
u115	3.209826e-10	u116	3.498565e-10	u117	5.679843e+02	u118	2.771999e-06	u119	1.757158e-01
u120	1.783598e-06	u121	5.259668e-10	u122	3.746471e-04	u123	2.113415e-06	u124	1.007405e+01
u125	5.344848e+01	u126	1.395878e+02	u127	5.080033e+02	u128	1.981534e+05	u129	8.094526e+05
u130	0	u131	5.222449e-03	u132	1.380312e+07	u133	1.210472e+08	u134	7.532941e-01
u135	8.460765e-01	u136	3.101805e-03	u137	7.688052e-02	u138	3.924507	u139	1.159746e-01
u140	2.106957e-01	u141	1.242754e+07	u142	3.635741e-02	u143	1.358760e+07	u144	4.299456e-02
u145	2.809204e+07	u146	4.586732e-02	u147	4.930859e-01	u148	8.859077	u149	4.495039e+06
u150	2.208813e-01	u151	4.651777e+06	u152	1.259095e-01	u153	8.571367e+02	u154	5.330477e+06
u155	1.940336e-01	u156	5.136096e-01	u157	5.328625e-01	u158	0	u159	0
u160	0	u161	0	u162	6.951241e+02	u163	4.685068e+02	u164	8.218390e+01
u165	1.208968e+02	u166	9.538816e+03	u167	3.820788e-07	u168	3.268536e+04	u169	5.697953e-10
u170	8.439507e+02	u171	4.546240e-09	u172	1.418792e+06	u173	1.333866e+06	u174	3.369624e-12
u175	4.492474e+03	u176	7.736602e-10	u177	3.181825e+02	u178	1.850638e+01	u179	1.866119e-01
u180	3.006009e-06	u181	1.994891e-11	u182	0	u183	0	u184	0
u185	0	u186	0	u187	0	u188	0	u189	0
u190	0	u191	0	u192	0	u193	0	u194	0
u195	0	u196	0	u197	0	u198	0	u199	0
u200	0	u201	1.884206	u202	0	u203	0	u204	2.268764e-04
u205	6.094237e+04	u206	3.000000e-02	u207	2.686606e+04	u208	5.890622	u209	2.927487e-03
u210	0	u211	1.616306	u212	6.691869e+07	u213	9.311524e-07	u214	1.575627e+06
u215	5.279387e+06	u216	1.404290e+02	u217	1.741800e+04	u218	4.828415e+06	u219	5.983991e-01
u220	1.879658e+01	u221	0	u222	0	u223	0	u224	0

And lastly, in Tables 3 and 4, we give a list of values for fixed parameters u and start values for parameters p to be varied, based on [6].

The measurements we use throughout the paper were taken from [15, Figure 7a]. We use the curve corresponding to the case of platelet poor plasma with 10 nM Thrombomodelin initial dosage. The measured output is

y⁢(t)=x29⁢(t)+1.2⋅x48⁢(t).

The total Thrombin is given as the weighted sum IIa+1.2⋅mIIa.

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Received: 2020-02-07

Revised: 2020-03-23

Published Online: 2020-04-15

Published in Print: 2020-06-01

A neural network assisted Metropolis adjusted Langevin algorithm

Abstract

A Appendix: A blood coagulation model

References

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