Machine learning structure preserving brackets for forecasting irreversible processes
Authors: Kookjin Lee
Nathaniel Trask
Panos Stinis
Venue: NeurIPS 2021 Poster (2021)
Keywords: structure preserving machine learning
neural odes
forecasting
dissipative systems

Abstract: Forecasting of time-series data requires imposition of inductive biases to obtain predictive extrapolation, and recent works have imposed Hamiltonian/Lagrangian form to preserve structure for systems with \emph{reversible} dynamics. In this work we present a novel parameterization of dissipative brackets from metriplectic dynamical systems appropriate for learning \emph{irreversible} dynamics with unknown a priori model form. The process learns generalized Casimirs for energy and entropy guaranteed to be conserved and nondecreasing, respectively. Furthermore, for the case of added thermal noise, we guarantee exact preservation of a fluctuation-dissipation theorem, ensuring thermodynamic consistency. We provide benchmarks for dissipative systems demonstrating learned dynamics are more robust and generalize better than either "black-box" or penalty-based approaches.

Machine learning structure preserving brackets for forecasting irreversible processes
Kookjin Lee School of Computing and Augmented Intelligence Arizona State University Tempe, AZ 85281
Nathaniel Trask
Center for Computing Research
Sandia National Laboratories A Albuquerque, NM 87123
natrask@sandia.ge
Panos Stinis
Pacific Northwest National Laboratory Richland, WA 99354
Abstract
Forecasting of time-series data requires imposition of inductive biases to obtain predictive extrapolation, and recent works have imposed Hamiltonian/Lagraan form to preserve structure for systems with reversible dynamics. In this work we present novel parameterization of dissipative brackets from metriplectic dynami- cal systems appropriate for learning irreversible dynamics with unknown a priori model form. The process learns generalized Casimirs for energy and entropy guar- anteed to be conserved and nondecreasing, respectively. Furthermore, for the case of added thermal noise, we guarantee exact preservation theorem, ensuring thermodynamic consistency. We provide benchmarks for dis- sipative systems demonstrating learned dynamics are more robust and generalize better than either "black-box" or penalty-based approaches.
coccuuationddisipaatio
1 Background and previous work
Modeling time-series data as a solution to dynamical system with learnable dynamics has been shown be effective in both data-driven modeling for physical systems and traditional machine learning (ML) tasks. Broadly, it has been observed that imposition of physics-based structure leads to more robust architectures which generalize well (T] On one end of the spectrum of inductive biases, universal differential equations (UDE) (2) assume an a priori known model form, thus imposing the strongest bias. On the other, neural ordinary differential equations (NODEs) assume a completely black-box model form with minimal bias.
Many recent approaches have turned structure preserving models of reversible dynamics obtain an inductive bias that lies in between 4 5
6 8. One may use black-box deep neural networks (DNNs) to learn energy of system with unknown model form, while the algebraic structure of Aamittonian/Lagraggiia dynamics provides a flow map which conserves energy. Typically, the learned flow map has symplectic structure so that phase space trajectories are conserved. In classifi- cation problems, this mitigates the vanishing/exploding gradient problem and improves accuracy [9]; in physics, this guarantees that extrapolated states physically realizable [TO].
Such approaches are only appropriate for reversible systems lacking friction or dissipation. In the physics literature, the theory of metriplectic dynamical systems provides generalization of the Poisson brackets of ieailltoninn/aaarangggnn mechanics which model not just conserved energy, but generalized Casimirs such entropy [2] Physical systems which can be cast in this framework
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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obtain a number of mimetic properties related to thermodynamic consistency: satisfaction of the first and second laws of thermodynamies and, for closed stochastic systems, fluctuation dissipation theorem (FDT) that guarantees thermal forcing is balanced exactly by dissipative forces in equilibrium [3]. This FDT property is particularly critical to analyzing rare events in molecular phenomena driven by thermal noise 14].
Classically, metriplectic systems are obtained by deriving a model from first principles and then observing that the system admits a requisite algebraic structure. While effective for a wide range of physical systems [3], the requisite first principles modeling may be restrictively complex for a general system, particularly for multiscale problems involving time history. In this work we reverse the process by assuming our time-series data has been generated by a metriplectic system and then inferring the requisite algebraic objects using a training strategy similar to that used in NODEs. This presents several technical challenges. First, dissipative systems typically have non-observable states (i.e. internal entropy or temperature) which may not be measured. Second, the metriplectic algebraic structure is particularly restrictive, requiring discovery of matrices with carefully designed null-spaces to separate reversible and irreversible components of the dynamics.
Finally, we note that this work is an important first step toward handling more complicated dissipative chaotic systems ubiquitous to science and engineering problems. For example, for chaotic systems the "butterfly effect" causes arbitrarily small perturbations in initial data to exponentially diverge, and it is only possible to provide long- term forecasting by learning a corresponding "strange attractor" whose latent dimension is governed by the dissipative structure 17. In reduced order-modeling, many have looked toward data-driven means of fitting dynamics to latent representations of solution space, with Hamiltonian structure particularly useful for finding long-time accurate surrogates 18, [9]. In this situation as well, structure peeeuurr-preeviingggg treatment of dissipation is critical to account for entropic/memory effects which emerge from coarse- gearrsiggiinnggggg ML is in robotic control (21). Again these models fail to account for friction due to wear, which is inevitable in realistic applications. There are also problems where we have access to time-series and no guaranteed way of modeling from first principles that can account for all the important mechanisms c.g. yystemiidentificttion in biology [22] (23. the study of cascading failures for realistic power grid models (24). social dynamics [25], and accounting for memory effects in empirical eigenfunction expansions of turbulent flows (26). All these cases can benefit by a structure preseevvnn method for identifying dynamics.
Anticipated impact:
20 Another success of reversible
2 Related work
Neural ordinary differential equations As noted, learning time-continuous dynamics in the form of a system of ODEs is an active topic with seminal works including [27 9 28 3
29 There have been many follow-up studies to enhance neural ODEs in different aspects, e.g., enhancing the expressivity
of neural ODEs by augmenting extra dimensions in state variables [30 checkpoint methods to
mitigate numerical instability and to enhance memory efficiency [31
32 33 allowing network
parameters to evolve over time together with hidden states (34] 35 and spectrally approximating
dynamics by using a set of orthogonal polynomials [36]. Applications of NODES for learning
complex physical processes (c.g.. turbulent flow) can be found in [37 38 39
Structure preserving neural networks A thorough accounting of works embedding structure- preservation into neural networks include pioneering works for Hamiltonian neural networks 4 40) followed by development of Lagragian neural networks 41 5 and neural networks that mimic the action of symplectic integrators 66 g More recently, there has been efforts to add physical invariance to learned dynamics models, e.g., time-reversal symmetry [42). Works pursuing related but distinct spatial- compaiibiiiyy related conservation structure other than geometric integration include: graph architectures with associated data-driven graph exterior calculus 43, solving optimization problems with conservation constraint in latent space [44], and adding conservation constraints as a penalty in training loss (5). The closest work to our approach is in [46), which proposed a time integrator that leverages the GENERIC (general equation for the nonequilibrium reversible-irreeersiee
coupling) formalism to impose the structure, but enforces the degeneracy condition as penalty terms in the training loss objective, We will provide results demonstrating that a penalty approach is insufficient to guarantee preservation of metriplectic structure.

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Theory and fundamentals
We consider the GENERIC formalism as particular metriplectic framework amenable to parame-
terization. Consider time series data D {(t, =tt,))) where the state a; = xxt,) ) C R has known initial condition 2o. In GENERIC, it is assumed that observable A(x) evolves under the gradient flow
AA +
(1)
where E and S denote generalized energy and entropy. {-:)) denotes Poisson bracket, and [:]]] denotes irreversible bracket. The Poisson bracket given terms of skew-symmetric Poisson matrix L and the irreversible bracket is given in terms of symmetric positive semi-definite friction matrix M,
dA AB dx
0A
3B M
{A, B)
and A. B
Ox
=
dx
dæ
A system governed by Eq.
1 is GENERIC system if the following degeneracy conditions hold
L, as = O, and aE
0.
(2)
dx
Taking æ in Eq. ① provides the evolution æ
dæ
0E
0S
+
M
dt
dx
dx
Remark 3.1 (Hamiltonian dynamics) For canonical coordinates æ =Iq. pl and canonical Pois- son matrix L
and M = O, Eq. 3 recovers Hamiltonian dynamics.
-1
Remark 3.2 (First and second laws of thermodynamics) Taking A = E and A = S, we obtain at = and ds or 2 O, respectively. This follows easily by application of the degeneracy conditions and noticing {A, A} = 0, [A 0.
Remark 3.3 (Fluctuation dissipation theorem) Introducing thermal noise to Eq. provides the stochastic differential equation (SDE)
dæ: -(1
OE Zx
OS Ox
+ M
KB
M dt / MWW
(4)
where V denotes the Cholesky factor of M, kg Boltzmann constant, and dW, Wiener process. The equilibrium statistics of this SDE reach a stationary distribution under appropriate conditions [13].
Parameterization of bracket structure
We now introduce parameterization of the dissipative and reversible brackets that exactly satisfies the degeneracy conditions described in Section 3 and review the penalty approach from [46] which imposes degeneracy conditions via soft constraints. Our approach is motivated by the work in [47 which we summarize in Sections 4. 4.3 For the remainder, we adopt the Einstein summation convention.
First, we parameterize the energy and the entropy as neural networks, i.e., E(x) x Ex(x) and S(x) (z)),, where and so are weights and biases for the neural networks E and S respectively.
4.1 Parameterizing skew-symmetric reversible dynamics
The reversible dynamics are characterized by a skew-symmetric Poisson bracket,
0A 0B as
{A, B) Eorr Dr Dz Ox,""

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where Easy is an skew-symmetric 3d tensor. To enforce the anti-symmetry exactly, we consider generic tensor East with learnable entries and apply the following skew-symmetriiztion trick.
1 Sagy 3! (Casy Eagg + E3yy - Esa + SnaB E,,8a)
The reversible part may then be written as {ar, F. = Easy Üs. dzg dxy are given by
Bn aE as and the reversible dynamics
do
0E as
Eagh
dt
Bx dxn
4.2 Parameterizing symmetric irreversible dynamics
Next, parameterize the irreversible dynamics via the bracket,
0A dE 0B 0E [A, B] So.8.pu ax dag dr dxv
where
Sa A D Apn Here, A and D) are skew-symmetric and symmetric positive semi-definite matrices, respectively, such that
A - Bal and Dmn Dum- Again, the skew-symmetry and the symmetric positive semi-definiteness can be achieved by the parameterization tricks
A - AT), and D= DD
where Á and D are matrices with learnable entries. Finally, the irreversible part may be written as as and the irreversible part the dynamics given by
[x. S) = So Uea Bng Dap Bay
dx
DE 0S dE
S. deg Dz, O.y
dt
irr
4.3 Degeneracy conditions
With the above parameterizations the degeneracy conditions described in Eq. 2 may be easily verified by direct calculation following the definition of the brackets and the symmetry/skew-synee symmmmt conditions.
Ox as
dæ. 0S as Easy Org Ox
0S as 0,
{a, S}
EaBy dra Ozg Ox.
Ox
8x
and
dx
DE
Ox 0E 0E 0E Sag Oxg Ox dz,
OE 0E 0E
x
M
Sa Oxa drg dx dx,
0.
dx
dæ
4.4 Alternative parameterization penalty-based method
An alternative strategy to incorporate GENERIC structure is to enforce the degeneracy condition
by soft penalty advocated in [46. In this approach, E, S, L, and M, may be approximated independently of each other. Again, E and S are parameterized as neural networks (F,g and S.), and T. and M are parameterized sieexsyymeeriiaaonsiymmeerzzziii of matrices and p with learnable entries as follows
L T -T-T") and Mp - pp
With this parameterization, penalty terms,
the degeneracy conditions are simply enforced by minimizing two We stress that this penalty will be enforced only to within
L dEg
and
M
as
Bar
dm
optimization eтror.
fe. where θ consists of learnable parameters, then Table 1 summarizes the components comprising fe for black-box NODE, the penalty-bass method, and GENERIC NODE (GNODE).
If we write system of neural ODEs as dr ăt

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Table 1: Model summary
NODE
Penalty
GNODE
fe
fe = fo
aEe fe = Li dæ
+ M
a5e
fe ar, E} + [x, S]
de
Esp and Sip (MIPs) Lr and Mp (2-tensor)
Eip and Sip (MLPs) e (3-tensor). A and D (2-tensor)
Components
black-box MI.P
=
= {Φ, (,,
= {Φ,, S, D)
Experiments
In this section, we assess the performance the three parameterizations of the ODE dynamics which apply progressively more stringent priors. We implement the algorithms in PYTHON 3.6.5, NUMPY 1.16.2, and PYTORCH 1.7.1 48.. For the time integrator, we use a PYTORCH implementation of differentiable ODE solvers, TorchDiffEq [3. All experiments are performed on MACBOOK PRo with 2.9 GHz i9 CPU and 32 GB memory.
5.1 Dataset and training
The states GENERIC systems may generally be partitioned between "observable" states (e.g.. position and momentum variables) denoted by x" and "non-observable" states (e.g., entropy, config- uration variables, ett) denoted by x"', i.e., æ = [x= _=" We assume that training data is only available for the observable states, with the non-observable states functioning as hidden variables dur- ing training. For each benchmark problem, we take as manufactured training data single trajectory of observable states obtained by integrating reference ODE with known GENERIC structure from known initial condition. We then split the sequence into three segments, [0, ttrain), (ftrain tvai], and (fval, &est] for training, validation, and test such that 0 < train < tval < &
We employ mini-batching to train all three considered architectures. Each mini-batch consists of multi- ple short sequences of length L whose initial conditions are randomly chosen from [0, feraiu]- To train "black-box" box" neural ODEs, we simply use a stochastic gradient descent (SGD) optimizer update the network weights and biases using the mini-batches on the observable states, {x² x + 2 1-1
As opposed the black-box neural ODEs, training the penalty-based approach and the GENERIC approach requires data to impose Mini-batch initial conditions (æt, #e+1. x-+ with x = [ x;]] where {x}} are unavailable. To address this is- sue, we propose a training strategy that alternately updates the model parameters and infers the non-observable states. We start with guess for the non-observable states. We then alternate between (1) updating the model parameters using SGD while fixing the current non-observable states and (2) updating the non-observable states by solving an initial value problem using the most recent model.
non-observable states, i.c.,
Algorithm 1: Neural ODE training
Initialize θ
for (i = 0; i Tmmax; = i do
Sample initial points (26 h where E(k) E [0, lerain L 1] for k ODESalvet fert..... f1.) for & = 1...., , No
I,...,N, No
Compute loss: L(x +m)
Update & via SGD
For ODESolve, we use the Dormand Prince method (dopri5) 449 with relative tolerance 10 and absolute tolerance 10-6 The loss function L measures the discrepancy between the ground truth states and approximate states via mean absolute errors, and the network weights and biases updated using Adamax [50] with an initial learning rate 0.01.
In the following, we test the proposed algorithms with two benchmark problems: damped nonlinear oscillator and two gas containers problems. Data for all considered benchmark problems can be found in [51).
5

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Algorithm 2: Penalty or GENERIC training
Initialize θ and {ar ... Ftriin
Construct a dataset 2; [ x.
T for i O,... Itrain
for (i = 0; i Tmaxx; i +)
Sample initial points (a2(()) 1--1 where {(k) E [0, terain - 1] for k 1, No. ODESolve(zza)) (.tii..dt) for 1,...; Nob
Compute loss: ((( +m)
Update via SGD
i mod Pupdate = then "Iya..,"ttrain = ODESolve(zo. Se
Update a dataset as x; [xp 0 T for = 0,..., Israin
10
5.2 Damped nonlinear oscillator
As first benchmark problem, we consider a damped nonlinear oscillator which exhibits a natural GENERIC structure:
P p dp kinnn))) YP dt mT
dS
74
(5)
m
where (q, p) denote the position and momentum of the particle, and S is the entropy of the surrounding thermal bath. The constant parameters m, 7, and T represent the mass of the particle, the damping rate, and the constant temperature of the thermal bath. The total energy of the GENERIC system is
p² 2m
F(g. P. S) = H(, p) + TS
k cos(g) + +TS,
where H(g.p) is the Hamiltonian of the particle (the sum of the kinetic and the potential energy).
In this benchmark problem, the observable states consist of the position and the momentum variables, i.., x [q.p]" _ We consider single non-observable variable, i.e., x = Now, our goal is to learn a system of ODEs that conforms the GENERIC structure described Section 4 and infer the non-observable variable via Algorithm 2
That is, for GNODE, we model Es and S.p to take
x= P. as an input.
For black-box NODEs, we have tested MLPs with combinations of (2, 3, 4) hidden layers with (5, 10, 15} neurons and observed the best result with an MLP with 4 hidden layers with neurons in each layer and hyperbolic tangent (Tanh) activation function. For the penalty-based approach, we have tested MLPs with combinations of {2, 3, 4} hidden layers with (5. 10, 15) and Tanh for parameterizing Eg and S. and observed the best results with MLPs with hidden layers with neurons in each layer. The x 3 learnable entries for La and Mo are considered. We add the penalty terms (see Section 4.4) that are weighted by 10 to the main loss objective. We have also tested 10-2 and 10-6 for weighting, which resulted in significant misfit in data inconsistency in physics (i.e., failure to enforce the degeneracy conditions). Lastly, for the GENERIC approach, we have tested MLPs with combinations of (0, 1, 2) layers with (5, 10) neurons for parameterizing Es and S.p and we have observed the best results with an MLP with I hidden layer with 10 neurons and Tanh for parameterizing Eg,, and a linear layer for parameterizing Spp. Then, we use 3 x 3 x 3 skew-symmetric tensor to parameterize ç, 3 x 3 skew-symmetric tensor to parameterize A, and 3 x tensor, d, to parameterize D, i.e., D ddd For initializing layers in MIPs, we the PYTORCH default uniform distribution and, for initializing learnable entries, we initialize them with unit normal distribution. We initialize the non-observable variable as xb = sę = te (i.c., setting it to be monotonically increasing) in Line 1 of Algorithm 2
The dataset consists of sequence of 180,000 timesteps with fanal 180 (in second) and step size At = 0.001. We then split the dataset into training, validation, and testing sets such that Atrai 20, fyal 40, and frest 180. Each mini-batch consists of N₂ 20 subsequences of length = 120. The maximum training step is set as 7mxx 30000 and the update is performed at every Nupdate 500 training steps (in Algorithm 2.
In the experiment, we consider m = = and y 0.01. The initial condition given as x = = 0, T) where the initial condition for the non-observable variable is arbitrarily set. Results

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of the comparison are given in Figure [ The results are obtained from 5 independent runs (i.e., 5 different random seeds). The plots of ds penalty formulation leads negative entropy production of large magnitude, while the GNODE approach enforces by construction ds both black-bor NODE and the penalty approach have large standard deviation in the predicted H. GNODE in contrast learns nearby entropy which consistently dissipates the correct amount of energy.
at reveal that the soft enforcement of constraints in the
dt 2 O. When extrapolating well beyond the training time interval,
5.3 Two gas containers
The second benchmark problem considers two (ideal) gas containers, separated by moving wall, exchanging heat and volume. Here, we are interested in the position and the momentum of the separating wall, i.e, x² [l. This problem possesses highly nonlinear expression for the entropy [51]
dq P di m
dp
2
E
E2
di
3
P
2
p
9N dS
1
ds²
9N2
1
1
dt
4E
E
Eg
dt
4E
E
E2
where (q, p) denote the position and momentum of the separating wall and S and S2 are the entropies of the two subsystems. The constants m denotes the mass the wall, 2hg is the total length of the two containers. Following [51], set Nkg 1, which fixes characteristic macroscopic unit of
Trajectory
penalty
GNODE
1.5
12.5
20
7.5
a 0.0
5.0
-20
-1.5
0.0
-2
180
180
7
(second)
(second)
(a) Trajectory
(b) 48
penalty
(c) ds
GNODE
Hlg, p) NODE
Hig. P) penalty
Hiq, GNODE
-1
-1
20 40
100
180
0
20 40
100 Isecondi
180
20 40
100 (second)
180
(seccond)
(d) NODE
(e) H penalty
(1) H GNNOD
Ground truth
Mean
Std Dev
Train
Validation
Figure 1: For the damped nonlinear oscillator, the physical entropy may be evaluated via the formula E H T.S. While all three methods fit training data reasonably well, NODE and the penalty approach rapidly deviate. An inspection of d5 for the penalty method shows that the soft penalty insufficient to ensure compatibility with the second law. GNODE is able to consistently learn an entropy 5 which closely tracks the physical entropy (E (E-H1) H)/T.

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Trajectory
db
penalty
GNODD
10
150
100
6
4
wwwwmm
-50
0
0.5
q
(second)
(second)
Trajectory
48
penalty
48
GNODE
Hlq. p) NODE
H(q.p) penalty
H(q. p) GNODE
5
10
5
10
30
5
10
30
(ssccond)
(second)
(second)
(d) H NODE
(e) H penalty
(l) GNODE
Ground truth
Approximate
Train
Validation
Figure 2: For the two gas cylinder problem the network must learn a significantly more complex entropy-energy relationship than in the damped oscillator. The results however are similar; GNODE provides remarkable improvement in forecasting due to its faithful reproduction of the second law of thermodynamics (top row, center + right).
entropy, and a 0.5. The internal energies of the two subsystems has the relationship with the associated entropies and volumes via the Sackur-Tetrode equation for ideal gases such that
S, In (E,
1,2,
Nkg where è is constant to ensure the argument of the logarithm dimensionless (set as 102.25). The total energy is given by
p2 2m
Elg. S1. S2) = H(q p) - F1 + F2
+ F1 + E2.
Again, the observable states consist of the position and the momentum variables, i.e., x= [g. PT We consider two non-observable variables, i.e., xx = [>1, 82]] This problem is more challenging as the dynamics of the observable variables strongly depends on the dynamics of the non-obesrvable variables. Now we train GNODE to learn system of ODEs that conforms the GENERIC structure described in Section 4 and infer the non-observed variables via Algorithm Again, for GNODE, model Eg and S.P to take x [g. p, 81, s2] as an input.
For black-box NODEs, we consider an MLP with hidden layers with 5 neurons in each layer and Tanh as nonlinearity. For the penalty- based approach, we consider MLPs with 3 hidden layers with neurons in each layer and Tanh for parameterizing Eg and Sy. and 3 x learnable entries for L and Mp- We add the penalty terms that are weighted by for the GENERIC approach, we consider an MLP with 2 hidden layer with 5 neurons and Tanh for parameterizing Eeg. and hidden layer with neurons and Tanh for parameterizing Sp- Then, we use 3 x 3 3 skew-symmetric tensor to parameterize &, 4 x skew-symmetric tensor to parameterize A.
to the main loss objective. Lastly,

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q analytic
Q- GNODE
p- -analytic
p
150
150
100
100
100
50
30
4.5 4.570042o-04
-.6827200-04
8.002106e-28
9.011495e-08
1.021312e422
1.040471-022
2:840518o-02
2,942702o-22
(a) q analytic
(b) GNODE
(c) analytic
(d) P GNODE
Figure 3: Distribution of q and p at länal 80 for the SDE The standard deviations of (g.p)
for baseline and GNODE are (9.9684 10-6 6, 1999 10-5) and (1.6436 x 10-6, respectively.
10-6,3.2977 ..3.2972 10 10-",,
and 4 tensor, d, to parameterize D, i... D dd For initializing layers in MLPs, the PYTORCH default uniform distribution and, for initializing learnable entries, we initialize them with unit normal distribution. The non-observable variables initialized 81,6 $2, 4g.
The dataset consists of a sequence of 30,000 timesteps with tinal = 30 seconds and step size At 0.001. We then split the dataset into training, validation, and testing subsets such that train 5, tval = 10, and frest = 30. Each mini-batch consists N₁ = 20 subsequences of length L = 40. The maximum training step set 7max 50000 and the update is performed every Nupdate = 500 training steps (in Algorithm
We m Lg and the initial condition is given as x 2,, Again, the initial condition for the non-observable variables are set arbitrarily. The results are depicted in Figure 2 and demonstrate similar results the damped oscillator. We depict the results from the best performing instances for each model out of five independent runs. Both NODE and the penalty method provide inaccurate forecasting beyond the training set, and the penalty method can be seen to generate negative entropy violating the second law of thermodynamics, while di for GNODE.
ds 0 holds strictly
5.4 Stochastic damped harmonic oscillator
We finally fit a GNODE model the system considered Section 5.2 and use the learned E, S, M and L in the right hand side of the SDE in 4 and compare as a baseline to using instead the analytical E, S. M, and from (5). This amounts to driving both the true system and data-driven dynamics with thermal noise which exactly balances the dissipation, and requires the FDT to hold to realize stationary statistics.
In this experiment, we consider the damped nonlinear oscillator with m = 10, and 0.16 and the same neural network architecture considered in Section [5.2] We use a sequence 80,000 timesteps (fiual = 80 and At 0.0001) to train the neural network. We use the same training strategy that is used in Section 5.2 (Algorithm 2); the only difference is that we use the mini-batch of size No- 40.
For Figure 3 we obtain statistics from solving 44 with linal = 80 of Maruyaaaa 52 with step size At 0.001. The mean of the resulting SDE solutions show substantial deviation, consistent with the fact that training is performed only on deterministic training data and not the SDE, which is compounded by nonlinearities in the data. However, the standard deviation of the distribution shows good agreement, suggesting that the FDT enforces thermodynamically consistent energy budget between dissipation and stochastic forcing.
6 Conclusions
We have constructed a generalization of structure preserving networks for reversible dynamics to handle dissipative processes. Unlike the reversible case, the bracket structure requires much more careful treatment of degeneracy conditions to ensure compatibility with the first and second laws of thermodynamics. Numerical examples show that our novel parameterization is able to provide non-decreasing entropy that translates improved robustness for out-of-distribution forecasting. We additionally show that exact treatment of dissipative processes allows introduction of thermal forcing

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which satisfies discrete FDT. To achieve themmodyyamically consistent equilibrium distributions in this setting, we have shown the degeneracy condition must be imposed exactly.
While this work establishes the value of imposing bracket structure for dissipative processes in terms of generalization, robustness, and physical realizability, the training approach applied here is applicable only to relatively small systems, restricting its application e.g. to learning dynamics of reduced-ordes models. Future work will focus on developing more scalable training strategies for learning ODEs of many variables.
7 Acknowledgements
Sandia National Laboratories is multimission laboratory managed and operated by National Technol- ogy and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's 's National Nuclear Security Administration under contract DE-NA0003530. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy the United States Government. SAND Number: SAND2021-7461 O.
We thank Chris Eldred for his recommendation to consider Ottingers bracket formulation of GENERIC. The work of N. Trask, and Stinis is supported by the U.S. Department of Energy, Office of Advanced Scientific Computing Research under the Collaboratory on Mathematics and Physics-Informel Learning Machines for Multiscale and Multiphysics Problems (PhILMs) project. N. Trask and K. Lee are supported by the Department of Energy early career program.
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12


Reviews:
[{"review_id": "d4eExylMV5j", "reviewer": "Reviewer_PQY3", "summary": "This work proposes a neural network architecture and training approach that leverage theories of dynamical systems for introducing a novel parametrization which splits the underlying time evolution dynamics into two parts – an energy conserving part, and as irreversible part. This allows one to use this framework for learning dynamics with dissipative states (for example, a damped oscillator). To accommodate systems with dissipative states that may not be directly measured, this work introduces a training scheme which alternates between system identification (assumes knowledge about both measured and non-measured states and tries to learn the underlying dynamics) and state estimation (assumes knowledge about the underlying dynamics and tries to estimate the non-measured states from the measured states). The proposed framework has been demonstrated on 3 example systems. To conclude, this paper addresses a very topical problem and the details have been provided with sufficient clarity. ", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "### Originality: \n\nThe overall idea is novel and very interesting. \n\nHowever, some relevant prior works are missing from the related work section. *Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning* (DeepDiffEq Worksho at ICLR’2020) and *OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle* (arXiv: 2009.02327) have focused on incorporating energy-dissipation into such physics-governed neural networks. The authors should properly refer to these papers and highlight the similarities/differences. *Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control* (ICLR’2020) and *Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints* (NeurIPS’2020) are also relevant references in this context. The authors may refer to *Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data* (arXiv:2012.02334) for comprehensive details about the various existing work that focus on incorporating Hamiltonian/Lagrangian-based inductive bias. \n\nAlso, Lagrangian-based neural networks precede Hamiltonian-based neural networks; *Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning* was published at ICLR’2019, almost 8 months before *Hamiltonian Neural Network* was presented at NeurIPS’2019 – this chronology contradicts the phrase “followed by” on line 68. \n\nLastly, references [21] and [36] refer to the same paper *Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning* – the authors should fix this.\n\n### Quality: \nThe experiments demonstrate that GNODE outperforms black-box Neural ODE as well as the penalty-based approach. However, in this context the update frequency for the unobserved state estimation appears to be a critical factor. Therefore, the authors should report on how the performance of GNODE and the penalty-based method varies with changes in that update frequency. Also, why $n_{update}$ was chosen to be the same for both of these methods? \n\n### Clarity: \nThis paper is very well-written which helps in conveying the message in an effective way. Also, it provides all the necessary details. \n\n### Significance: \nIn the last few years, a fast growing volume of work has focuses on incorporating Hamiltonian of Lagrangian formulation based inductive bias while learning dynamics of data. However, most of these papers (except *Dissipative-SymODEN* and *OnsagerNet*) encode *energy conservation* in the architecture of neural networks. This paper expands that line of work by incorporating energy-dissipation into a such energy conserving neural networks. As almost all real-world physical systems have some-form of energy dissipation, this reviewer believes that the proposed approach would be relevant for many practical applications.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors have adequately addressed the limitations of the submitted work.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "Wi1jL1Mj0JG", "reviewer": "Reviewer_DqnY", "summary": "The paper introduces a new framework for learning neural ODEs obeying metriplectic dynamics. These differ from previous work on Hamiltonian dynamics in that it includes both conservative and dissipative terms guaranteeing that system entropy is non-decreasing. Dissapative dynamics are extremely common in real-world systems and so this is an important inductive bias to be able to account for. The authors compare their proposed method with black box and penalty based approaches and find that their proposed method outperforms both.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The writing and motivation in the paper are very strong, though there are a few terms missing, namely the $s_i$ in the experiments. They seem to be specific entropy, but I didn't catch a definition in the paper. Overall on this front, the motivation is clear and the approach taken seems to be well founded. To my knowledge, the approach taken is unique though it seems well-grounded in previous work and a natural extension from previous work on learning Hamiltonian dynamics. \n\nThe main weakness of the paper is in the experiments and presentation of the results. The method shows promise and it's positive to see that H and S do seem to enjoy the proposed theoretical benefits in practice. There are a few aspects of the experiments that seem under-explained and perhaps overfit to the experiments:\n- D is defined explicitly to be rank 1. This is not a requirement in the general framework and the matrix is too small for this to seem motivated by storage costs, but it seems like it would reduce the influence of the disappative term. It's hard to tell if this is theoretically or experimentally motivated. \n- The configurations of the networks seem to vary significantly and this isn't explained. All tests seem to be done on small systems with correspondingly small networks. While the additional parameters of the penalty and GNODE methods may motivate smaller networks for fair comparison, the significant differences make it hard to tell if the better fit is due to the method or if it's just that neural ODEs are hard to train and the smaller networks led to better fits on relatively simple dynamics. \n- The penalty comparison does not feel very robust. One value of the multipliers is reported without indication as to how they were chosen. It seems odd that the penalty method would completely underperform the black box approach as is seen in figure one and without the error bars, I'd guess it was a bad optimization run. Due to the small number of runs, it still seems possible that they were influence by an outlier.\n- These seem like one particle systems, so if the target of interest is mapping the trajectory, then I'd like to see results reported on the discrepency between the true and targeted trajectories and not just H as the black box method is not trained to optimize performance on H. \n\nFinally, there were also a few discrepancies between what was stated in the checklist and what appeared in the paper. First, I do not think it is reasonable to take credit for releasing code when the code is not included in the supplementary material. If the code isn't included in the submission, then the answer to \"did we release the code?\" is no regardless of future intentions. Error bars are present, but wide and it's unclear if these are the extremes or a number of standard deviations. These are also smaller experiments, so it seems as though more than 5 runs should be feasible particularly when the deviation seems to be so large. \n\nOverall, the paper is solid and proposed a promising direction, but in it's current version, the experiments are hard to put significant stock into. It would make for a stronger paper if larger systems were explored or more consistent configurations were used. Additional comparisons would also be valuable as vanilla neural ODEs are not a very strong baseline. It would also be interesting to see how the model preserves on non-metriplectic systems (Hamiltonian or unknown systems from collected data) to see if the technique is capable of adapting in those settings (ie, it should be able to represent Hamiltonian dynamics, does it outperform black box ODEs on those tasks?) though obviously it would not be expected to outperform methods designed for those settings.\n\nEDIT: Updated score based on author comments. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Limitations and societal impact are adequately addressed.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "5fdKkiqJceh", "reviewer": "Reviewer_hnow", "summary": "The authors introduce a hard-constrained, parameterized GENERIC framework for learning irreversible (dissipative) dynamics. To guarantee total energy conservation and entropy production, the authors use reformulation techniques for Poisson (reversible) and irreversible brackets that automatically satisfy the degeneracy conditions for GENERIC. As a result, the authors’ model can learn various dynamical systems more reliably and plausibly, compared to the baselines such as conventional neural ODE or soft-constrained GENERIC (that learns the degeneracy condition via penalty loss.) The authors validate their model for various benchmarks including damped nonlinear oscillators, two gas containers, and stochastic damped oscillators. ", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: The main contribution of this paper is that the proposed model guarantees the degeneracy conditions (that should be satisfied for obeying the laws of thermodynamics) automatically while the prior work only guides such conditions via regularizers. This can be achieved by reformulating the reversible and irreversible brackets (Section 4.1 and Section 4.2). However, these reformulations are identical to the work by Oettinger ([42] in the paper). The authors pick some appropriate formulas from Oettinger’s paper, and then directly parameterize the formulas to enable them to learn. As a result, this paper can have originality from the perspective of bridging/combination of well-known techniques, but not definitely novel. \n\nOne interesting point is that the authors distinguish the observable (position and momentum) and non-observable (entropy, …) states, and then learn the non-observables unsupervisely. I think this is an original point of this paper that might be important for real-world problems. \n\nOverall, I think this paper has moderate novelty/originality.\n\nQuality: I think this paper is technically adequate. GENERIC framework and modified reversible/irreversible brackets are principled structures to learn and extrapolate dissipative dynamical systems. Experimental results seem to be solid, except for some minor concerns (see below comments 1 and 2).\n\nClarity: This paper is extremely well-written and easy to follow. The motivation is clearly presented. Prior works are well summarized (while some works are missing. See below comment 3). The preliminary and specific techniques are appropriately introduced. The experimental session sufficiently provides details on settings and hyper-parameters. Figures effectively show the experimental result.\n\nSignificance: Learning irreversible dynamical systems is definitely important for solving real-world problems. This paper provides a solid contribution to such important problems. \n\nI think this paper is good, especially in terms of clarity. However, the lack of originality and some concerns (see below) prevent me from recommending clear acceptance for this paper. My current evaluation of this paper is borderline.\n\nSome concerns on the paper:\n\n1. The authors used $10^{-4}$ penalty loss weight when they train the penalty-based GENERIC (line 191 and line 233). It seems $10^{-4}$ is too small to learn the degeneracy conditions. Why did you use such small penalty strength? What if one increases the penalty weight, e.g., $10^{-1}$ or 1?\n\n2. For the two gas containers experiment (Section 5.2), the error bar / standard deviation is not reported, unlike the damped oscillator experiment (Section 5.1). Why?\n\n3. There are some uncited prior works that try to model dissipative or chaotic dynamics, such as [1] Y. D. Zhong, B. Dey, and A. Chakraborty. Dissipative SymODEN: Encoding Hamiltonian dynamics with dissipation and control into deep learning. ICLR Workshop on Integration of Deep Neural Models and Differential Equations, 2020, [2] K. Course, T. Evans, and P. Nair. Weak form generalized Hamiltonian learning. NeurIPS, 2020.\n\n***\nPost-rebuttal: After reading the authors' response, now I think the paper’s contribution outweighs my initial concerns (especially regarding the originality). I increased my review score accordingly.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors mention the limitation of this work, the scalable issue, in the conclusion.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This is a very good paper and a clear accept. All reviewers agree that there is a novel methodological contribution that extends neural ODE approaches to irreversible dynamical systems. The parametrization of brackets cleverly preserve structures of the dynamical system, leading to exact preservation of thermodynamical consistency and better generalization. The numerical experiments are good, but can be further improved. The paper is written nicely and the main ideas are explained well. 
Decision: Accept (Poster)