Certified data-driven physics-informed greedy
auto-encoder simulator
Xiaolong He
Department of Structural Engineering
University of California, San Diego
La Jolla, CA, 92093
xih251@ucsd.edu

Youngsoo Choi
Center for Applied Scientific Computing
Lawrence Livermore National Laboratory
Livermore, CA, 94550
choi15@llnl.gov

William D. Fries
Applied Mathematics
School of Mathematical Sciences
University of Arizona
Tucson, AZ, 85721
frieswd@math.arizona.edu

Jonathan L. Belof
Materials Science Division
Physical and Life Science Directorate
Lawrence Livermore National Laboratory
Livermore, CA, 94550
belof1@llnl.gov

Jiun-Shyan Chen
Department of Structural Engineering
University of California, San Diego
La Jolla, CA, 92093
js-chen@ucsd.edu

Abstract
A parametric adaptive greedy Latent Space Dynamics Identification (gLaSDI)
framework is developed for accurate, efficient, and certified data-driven physicsinformed greedy auto-encoder simulators of high-dimensional nonlinear dynamicalsystems. In the proposed framework, an auto-encoder and dynamics identification
models are trained interactively to discover intrinsic and simple latent-space dynamics. To effectively explore the parameter space for optimal model performance,
an adaptive greedy sampling algorithm integrated with a physics-informed error
indicator is introduced to search for optimal training samples on the fly, outperforming the conventional predefined uniform sampling. Further, an efficient k-nearest
neighbor convex interpolation scheme is employed to exploit local latent-space
dynamics for improved predictability. Numerical results demonstrate that the proposed method achieves 121 to 2,658 × speed-up with 1 to 5% relative errors forradial advection and 2D Burgers dynamical problems.

1

Introduction

Physical simulations have played an increasingly significant role in the developments of engineering,science, and technology. However, high-fidelity forward physical simulations can be computationallyintractable even with high-performance computing, prohibiting their applications to problems that
require a large number of forward simulations [1–9]. In recent years, several reduced-order models(ROMs) have been integrated with latent-space learning algorithms [10–13]. However, the latentspace dynamics models of these methods are often complex with limited interpretability. Many
methods have been developed for the identification of interpretable governing laws from data,
including symbolic regression that searches both parameters and governing equations simultaneously
Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

[14–17], parametric models that fit parameters to equations of a given form, such as the sparse
identification of nonlinear dynamics (SINDy) [18, 19], and operator inference [20–22]. Champion,
et al. [19] applied an auto-encoder for nonlinear projection and SINDy to identify simple ordinary
differential equations (ODEs) that govern the latent-space dynamics. The auto-encoder and the
SINDy model were trained interactively to achieve simple latent-space dynamics. However, theproposed SINDy-autoencoder method is not parameterized and generalizable. Many non-intrusive
ROMs have been developed based on POD-based linear projection with latent-space dynamics
captured by polynomials through operator inference [20–25]. Due to the limitation of the POD-based
linear projection, these non-intrusive ROMs have difficulties with advection-dominated problems. To
address this challenge, Fries et al. [26] proposed a parametric latent-space dynamics identification(LaSDI) framework, in which an auto-encoder was applied for nonlinear projection and a set of local
dynamics identification (DI) models were introduced to identify local latent-space dynamics. The
LaSDI framework can be viewed as a generalization of the aforementioned non-intrusive ROMs built
upon latent-space dynamics identification since it allows linear or nonlinear projection and enables
latent-space dynamics to be captured by flexible DI models based on general nonlinear functions.
However, the lack of interaction between the auto-encoder and the DI models due to a sequentialtraining procedure leads to a strong dependency on the complexity and quality of the latent-space
dynamics on the auto-encoder architecture, which could pose challenges to the subsequent dynamics
identification and thus affect the model performances. Most importantly, all these approaches rely on
predefined training samples, such as uniform or Latin hypercube sampling, which may not be optimalfor the best model performance in the prescribed parameter space.
To address these issues, we present a gLaSDI framework for accurate and efficient physics-informed
data-driven reduced-order modeling. For an optimal sampling of the parameter space, an adaptive
greedy sampling algorithm integrated with a physics-informed residual-based error indicator isintroduced to search for optimal and minimal training samples on the fly. The proposed gLaSDI
framework contains an auto-encoder for nonlinear projection to discover intrinsic latent representations and a set of local DI models to capture local latent-space dynamics, which is further exploitedby an efficient k-nearest neighbors (k-NN) convex interpolation scheme. The auto-encoder training
and dynamics identification take place interactively to achieve an optimal identification of simple
latent-space dynamics. Our numerical experiments demonstrate the effectiveness of the proposed
gLaSDI framework and a considerable speed-up is achieved.

2

Greedy Latent Space Dynamics Identification (gLaSDI)

Let us consider a system of ODEs characterizing a parameterized nonlinear dynamical system, which
can also be considered as a semi-discretized equation of a system of partial differential equations(PDEs) with a spatial domain Ω,
du(t; µ)= f (u, t; µ),dtt ∈ [0, T ],with u(0; µ) = u0 (µ),(1)

where T is the final time; µ is the parameter in a parameter domain D; u ∈ RNu is the parameterized
time-dependent solution to the dynamical system with an initial state u0 ; f denotes the velocity. With
the implicit backward Euler time integrator, an approximate solution to Eq. (1) can be obtainedby solving the nonlinear system of equations un = un−1 + ∆tfn , where un := u(tn ; µ), andfn := f (u(tn ; µ), tn ; µ). The residual function is defined asr(un ; un−1 , µ) = un − un−1 − ∆tfn .

(2)

The physics-informed adaptive greedy sampling is performed on a discrete parameter space (Dh ⊆ D)
and D ⊆ Dh denotes a set of Nµ selected training sample points. Let’s consider the i-th training(i)sample point µ(i) ∈ Dh and un as the associated solution at the n-th time step of the dynamicalsystem in Eq. (1). The solutions at all time steps are arranged in a snapshot matrix denoted as U(i) =(i)(i)[u0 , ..., uNt ]. The reconstruction loss of the auto-encoder is defined as Lrecon := ||U − Û||2L2 ,
N
N
(i)
(i)
(i)
(i)
= ϕd (zn ), and
(i)
(i)
(i)
zn = ϕe (un ); ϕe is an encoder and ϕd is a decoder; zn ∈ RNz denotes the latent variable withµµwhere U = {U(j) }j=1
and Û = {Û(j) }j=1
, Û(i) = [û0 , ..., ûNt ], ûnNz ≪ Nu .

2

(i)

(i)

Given the latent variable in the discrete form Z(i) = [z0 , ..., zNt ] obtained from the encoder, thegoverning equation of its dynamics is approximated by a DI model, ψ DI , based on a user-defined
library of basis functions Θ(Z(i)T ), e.g., polynomial, trigonometric, and exponential functions, such˙that Ż(i)T ≈ Ẑ(i)T = Θ(Z(i)T )Ξ(i) , where Ξ(i) is a coefficient matrix. See [26] for more details.
To identify simple and smooth latent-space dynamics, two additional loss functions, Lż and Lu̇ ,are constructed to enable interactive training of the auto-encoder and the DI model (interactive
˙
Autoencoder-DI training), as illustrated in Fig. 1, where Lż := ||Ż − Ẑ||2L2 ensures the consistency of˙
the latent-space dynamics gradients, and Lu̇ := ||U̇ − Û||2L2 ensures the consistency of the physical˙dynamics gradients. Note that Ż and Û can be computed through the chain rule, as illustrated inFig. 1. The local DI models are considered to be point-wise (see more details in [26]), which means
each local DI model is associated with a distinct sampling point in the parameter space. Hence, eachsampling point has an associated DI coefficient matrix.

Figure 1: Schematics of the gLaSDI algorithm. The black square boxes in the right panel indicate thesampled parameter points in a 2D parameter space.
It is expected that the latent-space dynamics within a small neighborhood of a parameter point are
similar. To exploit the local latent-space dynamics captured by the local DI models for enhanced
parameterization and efficiency, a k-NN convexity-preserving partition-of-unity interpolation scheme
based on Shepard function [27] or inverse distance weighting is employed. The associated DI
coefficient matrix of a testing parameter µ ∈ D is obtained by a convex interpolation of coefficientPmatrices of its k-nearest neighbors (existing sampling points), Ξinterp = i∈Nk (µ) Ψ(i) (µ)Ξ(i) .
The k-nearest neighbors are selected based on the Mahalanobis distance between the testing and thetraining (existing) parameters. The interpolation functions Ψ(i) (µ) with an inverse-distance kernel
function is employed, preserving convexity and a partition of unity.
To address the issues of parameter dependency of local latent-space dynamics efficiently and effectively, a physics-informed adaptive greedy sampling procedure is introduced to construct an optimalNµtraining database DB = {U(i) }i=1
on the fly during off-line interactive autoencoder-DI training, as
Nµillustrated in Fig. 1, which corresponds to a set of parameters D = {µ(i) }i=1
sampled from Dh withhNµ <= |D |. The training database is first initialized with a small set of parameters located, e.g.,
at the corners or at the center of the parameter space. To enhance sampling reliability and quality,
the model training is performed for a number of epochs before every greedy sampling, as illustrated
3

in Fig. 1. To expedite the sampling procedure, a random-subset evaluation strategy is adopted. At
the v-th sampling iteration, a small set of candidate parameters, Dsub ⊆ Dh (Dsub ∩ Dv−1 = ∅), arerandomly selected and the parameter µ∗ that maximizes a residual-based error indicator, eres (µ),
is selected to update Dv = {Dv−1 , µ∗ } and DB v = {DB v−1 , U∗ }. The residual error indicator isdefined as
NtsX
1||r(ûn ; ûn−1 , µ)||L2 ,(3)eres =
Nts + 1 n=0
where the residual r(ûn ; ûn−1 , µ) is physics-informed and obtained from Eq. (2) that relies on
discretized governing equations; Nts ≈ 0.1Nt to enhance the computational efficiency. The residual
error indicator is calculated based on only predictions and positively correlated with the maximum relative error, as demonstrated in Fig. 1. The maximum residual error (eresv ) of the trainmax
ing samples at the v-th iteration can be used to estimate the maximum relative error
 (ev ) inres
res
the parameter space based on a linear least-square fit between Ev = {eÛ(µ) }µ∈Dv and

max
max
Ev
= {e
U(µ), Û(µ) }µ∈Dv . The greedy sampling procedure continues until a prescribed
number of sampled points or a target error tolerance is reached. The trained gLaSDI model canthen be applied to efficiently predict dynamical solutions given a testing parameter in the prescribedparameter space.

3

Results and Discussion

The performance of gLaSDI is demonstrated by solving a two-dimensional parameterized radial
advection problem and compared with that of LaSDI [26] (without autoencoder-DI interactive training
and adaptive greedy sampling):∂u+ v · ∇u = 0, Ω = [−1, 1] × [−1, 1], t ∈ [0, 3],(4)∂twith a boundary condition u(x, t; µ) = 0 on ∂Ω and an initial condition u(x, 0; µ) =sin(w1 x1 )sin(w2 x2 ) parameterized by µ = {w1 , w2 }, where v = π2 d[x2 , −x1 ]T denotes the fluid
velocity with d = (1 − x21 )2 (1 − x22 )2 . The spatial domain is discretized by first-order periodic
square finite elements constructed on a uniform grid of 96 × 96 discrete points. The fourth-order
Runge-Kutta explicit time integrator with a uniform time step of ∆t = 0.01 is employed. A discrete
parameter space Dh is constituted by the parameters of the initial condition, including w1 ∈ [1.5, 1.8]
and w2 ∈ [2.0, 2.3], each with 21 evenly distributed discrete points in the respective parameter range.
The distributions of the solution field and its gradient at a few time steps for the parameter case
(w1 = 1.5, w2 = 2.0) are shown in Fig. 1. The gLaSDI model consits of Linear DI models andan auto-encoder that has an architecture of 9,216-100-3-100-9,216, with the numbers denoting thenumber of neurons in each hidden layer, and the latent dimension as 3. The training and testing are
performed on a NVIDIA V100 (Volta) GPU with 3,168 NVIDIA CUDA Cores and 64 GB GDDR5GPU Memory. The open-source TensorFlow library [28] and the Adam optimizer [29] are employed
for model training. The gLaSDI training is performed until the total number of sampled parameter
points reaches 25. A LaSDI model with the same architecture of the auto-encoder and DI models
is trained using 25 predefined training points uniformly distributed in a 5 × 5 grid in the parameter
space.
Owning to the interactive autoencoder-DI training, gLaSDI is able to capture simpler and smootherlatent-space dynamics than LaSDI that has sequential and decoupled training of the auto-encoder
and DI models, as shown in Figs. 2(a-b). A better agreement between the latent-space dynamics
predicted by the encoder and the DI models is achieved by gLaSDI. Figs. 2(c-d) show that gLaSDI
achieves a maximum relative error of 2.0% in the whole parameter space, lower than 5.4% of LaSDI.
When the parameter space is enlarged to w1 ∈ [1.5, 2.0] and w2 ∈ [2.0, 2.5], gLaSDI maintains ahigh accuracy with a maximum relative error of 3.3%, much lower than 24% of LaSDI, as shown in
Figs. 3(c-d). The results demonstrate that given the same number of training parameters, gLaSDI
with the physics-informed adaptive and sparse sampling can intelligently identify the optimal training
parameter points to achieve higher accuracy than LaSDI based on predefined uniformly distributed
training parameters. It is interesting to note that changing the parameter space affects the distributionof gLaSDI sampling. Compared with the high-fidelity simulation based on MFEM [30], the gLaSDI
model achieves 121× speed-up.
4

Figure 2: Comparison between LaSDI and gLaSDI with the same model architecture for the radial
advection problem with parameters w1 ∈ [1.5, 1.8] and w2 ∈ [2.0, 2.3]. The number on each box
denotes the maximum relative error of the associated parameter case. The black square boxes indicate
the sampled training parameter points.

Figure 3: Comparison between LaSDI and gLaSDI with the same model architecture for the radial
advection problem with parameters w1 ∈ [1.5, 2.0] and w2 ∈ [2.0, 2.5]. The number on each box
denotes the maximum relative error of the associated parameter case. The black square boxes indicate
the sampled training parameter points.
A potential limitation is the residual computation, which might not be straight-forward, for example,
if the source code of the full-order model is not available. Then, the physics information needs to bepost-processed with the gLaSDI solution, which might not be accurate. Another limitation is that thetraining of the auto-encoder can be computationally expensive as the data size becomes large, which
can be improved by applying the convolutional auto-encoder. The training efficiency can potentiallybe further improved by a combination of pre-training and re-training. The proposed gLaSDI
framework is general and not restricted by the use of auto-encoders and DI models. Depending on
applications, linear compression techniques with much lower training cost could be employed. Further,other system identification techniques or operator learning algorithms could be applied to identify
latent-space dynamics. The auto-encoder architecture can be optimized to maximize generalization
performance by integrating automatic neural architecture search into the proposed framework. The
parameterization in this study only considers the parameters from the initial conditions of the problems.
The proposed framework can be easily extended to account for other parameterization types, suchas of material properties, which will be useful for inverse problems. All the codes to regenerate theresults in this paper can be found in the GitHub repository (https://github.com/LLNL/libROM/
tree/gLaSDI/python/gLaSDI). The following existing assets have been used: SindyAutoencoderwith MIT license (https://github.com/kpchamp/SindyAutoencoders) and the MFEM librarywith BSD-3-Clause (https://github.com/mfem/mfem).

5

4

Broader impact

This paper introduces a novel machine learning enhanced physics-guided data-driven reduced-ordermodeling strategy with adaptive sampling to accelerate high-dimensional physical simulations.
The proposed gLaSDI framework is expected to have broad impacts on the computational science
community and application potentials in a wide range of engineering and scientific domains. There isno negative consequence on ethics and society in this work.

Appendix
An additional experiment and results are presented to further demonstrate the effectiveness of the
proposed framework. A two-dimensional (2D) parameterized inviscid Burgers equation is considered∂u1+ u · ∇u =∆u, Ω = [−3, 3] × [−3, 3], t ∈ [0, 1],
(5)
∂t
Re
||x||2

with a boundary condition u(x, t; µ) = 0 on ∂Ω, an initial condition u(x, 0; µ) = ae− w2parameterized by µ = {a, w}, and a Reynolds number Re = 10, 000. A uniform spatial discretization
with 60 × 60 discrete points is applied. The first order spatial derivative and the diffusion term areapproximated by the backward difference scheme and the central difference scheme, respectively.
The full-order model solutions are obtained by solving the semi-discretized system with the implicit
backward Euler time integrator and a uniform time step of ∆t = 1/200. A discrete parameter space
Dh is constituted by the parameters of the initial condition, including the width, w ∈ [0.9, 1.1],
and the amplitude, a ∈ [0.7, 0.9], each with 21 evenly distributed discrete points in the respective
parameter range. The solution fields of the first velocity component at different time steps for theparameter case (a = 0.7, w = 0.9) are shown in Fig. 4(a). The auto-encoder with an architecture of
7,200-100-5-100-7,200 (latent dimension as 5) and quadratic DI models are employed. The gLaSDI
training is performed until the total number of sampled parameter points reaches 36. A LaSDI modelwith the same architecture of the auto-encoder and DI models is trained using 36 predefined training
points uniformly distributed in a 6 × 6 grid for the parameter space.
Figs. 4(b-c) show that gLaSDI identifies much simpler and smoother latent-space dynamics than
LaSDI, with a better agreement between the encoder and the DI predictions, which is attributed by theinteractive autoencoder-DI training of gLaSDI, as illustrated in Fig. 1 and described in Section 2. Figs.
4(d-e) show that gLaSDI achieves the maximum relative error of 5% in the whole parameter space,much lower than 255% of LaSDI. The poor accuracy of LaSDI could be caused by the deviation
between the DI predicted dynamics and the encoder predicted dynamics. It is also observed thatgLaSDI tends to have denser sampling in the lower range of the parameter space. This demonstrates
the importance of the physics-informed greedy sampling procedure. Compared with the high-fidelity
simulation based on an in-house Python code, the gLaSDI model achieves 871× speed-up.
When the latent dimension is reduced from 5 to 3 and the polynomial order of the DI models is
reduced from quadratic to linear, gLaSDI learns simpler latent-space dynamics, as shown in Fig. 5(b),and maintains a high accuracy with a maximum relative error of 4.6%, much lower than 22% of
LaSDI, as shown in Figs. 5(c-d). Furthermore, simplifying the latent-space dynamics enhances the
reduced-order modeling efficiency. Compared with the high-fidelity simulation, the gLaSDI model
achieves 2,658× speed-up, which is 3.05 times the speed-up achieved by the gLaSDI model with alatent dimension of 5 and quadratic DI models.

Acknowledgements
This work was performed at Lawrence Livermore National Laboratory and partially funded by
two LDRDs (21-FS-042 and 21-SI-006). Lawrence Livermore National Laboratory is operated by
Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear
Security Administration under Contract DE-AC52-07NA27344.

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