Thermodynamically Consistent Latent Dynamics Identification for Parametric Systems
Authors: Xiaolong He
Yeonjong Shin
Anthony Gruber
Sohyeon Jung
Kookjin Lee
Youngsoo Choi
Venue: Accepted by TMLR ()

Abstract: We propose an efficient thermodynamics-informed latent space dynamics identification (tLaSDI) framework for the reduced-order modeling of parametric nonlinear dynamical systems. This framework integrates autoencoders for dimensionality reduction with the newly developed parametric GENERIC formalism-informed neural networks (pGFINNs), which enable efficient learning of parametric latent dynamics while preserving key thermodynamic principles, such as free energy conservation and entropy generation, across the parameter space. To further enhance model performance, a physics-informed active learning strategy is incorporated, leveraging a greedy, residual-based error indicator to adaptively sample informative training data, outperforming uniform sampling at equivalent computational cost. Numerical experiments on the Burgers' equation and the 1D/1V Vlasov-Poisson equation demonstrate that the proposed method achieves up to 2,495x speed-up over the full-order numerical baseline with 1-3% relative errors, as well as significant reductions in training (50-90%) and inference (57-61%) cost.
Moreover, the learned latent space dynamics reveal the underlying thermodynamic behavior of the system, offering valuable insights into the physical-space dynamics. Code is available at the repository: https://github.com/xiaolong7/pGFINN-tLaSDI.

Published in Transactions on Machine Learning Research (02/2026)
Thermodynamically Consistent Latent Dynamics Identifica tion for Parametric Systems
Xiaolong He
#iolongghee@agnopsye..omm
Synopsys Inc.
Yeonjong Shin
yeonjong shintmncuuedd
Department of Mathematics North Carolina State Universily
Anthony Gruber
Center for Computing Research Sandio National Laboratories
Sohycon Jung
sjung55 Bang5550sss..o edu
School of Mathematics and Statistical Sciences Arizona State University
Kookjin Lee
kookjjn.ee@@asu.c edu
School of Computing and Augmented Intelligence Arizona State University
Youngsoo Choi
choi15@llni
Center for Applied Scientific Compuling Lawrence Livermore National Laboratory
Reviewed on OpenReview: https: // opwp://veee. net/ forum? id= Qy3oLpRep
Abstract
We propose efficient thermoddynmiie--noouuddd latent space dynamics identification (ILaSDI) framework for the reduced-order modeling of parametric nonlinear dynamical systems. This framework integrates autoencoders for dimensionality reduction with the newly developed parametric GENERIC formalismninformeed neural networks (PGFINNs), which enable elficient learning of parametric latent dynamics while preserving key ther- modynamic principles, such as free energy conservation and entropy generation, across the parameter space. To further enhance model performance, a physics-informed active learning strategy is incorporated, leveraging greedy, residual-based error indicator to adaptively sample informative training data, outperforming uniform sampling at equiva- lent computational cost. Numerical experiments on the Burgers' equation and the 1D/IV Vlasov-Poisson equation demonstrate that the proposed method achieves up to 2, 495x speed-up over the full-order numerical baseline with 1-3% relative errors, as well as signif- icant reductions training (50-90%) and inference (57-61%) cost. Moreover, the learned latent space dynamics reveal the underlying thermodynamic behavior of the system, offer- ing valuable insights into the physical-space dynamics. Code is available at the repository: htt.a/b...o...toooooooo///////a0
1 Introduction
Understanding and simulating nonlinear dynamical systems is fundamental to advancing numerous fields in science and engineering, from physics (Oberkampf & Trucano) 2002 Thijssen 2007) and electromagnetics

---

Published in Transactions on Machine Learning Research (02/2026)
Bondeson et al. 2012: to material sciences Lee
2016 and digital twin technologies (Jones et al. 2020 Liu
et al. 2021 However, high-fidelity numerical models for such systems are often computationally prohibitive, especially in real-time or many-query settings such as optimization (Sigmund & Maute] 2013 White et al. 2020 uncertainty quantification (Galbally et al. 2010 Abdar et al. 2021]. J. or digital twin deployment (Jones et al.
2020 Liu et al., 2021). Reduced-order models (ROMs) offer a promising solution to this challenge
through their identification of low-dimensional representations that capture the essential dynamics of the system.
Traditional ROMs based on linear projection techniques, such proper orthogonal decomposition (POD) [Berkooz et al. 1993) and reduced basis methods (Patera & Rozza 2007 have demonstrated great promise, but typically require direct access to high-fidelity numerical solvers (McBane & Choil 2021). Additionally, they often struggle with advection-dominated or highly nonlinear systems, characterized by the slow decay of their Kolmogorov n-width, which quantifies the best-case approximability of the system state using a lincar subspace of dimension n (Frics et al., 2022). To address these limitations, nonlinear manifold learning via autoencoders (Hinton & Salakhutdinov 2006) has been explored, offering enhanced expressiveness for
capturing complex solution manifolds (Fulton et al. 2019 Lee & Carlberg 2020 Kim et al. 2022 Maulik
ct al.
2021 Lee & Parish 2021 at the cost of greater computational complexity. Recent advancements
further integrate equation learning methods (Schmidt & Lipson 2009 Brunton et al.
2016 Peherstorfer &
Willcox
2016:
Messenger & Bortz 2021b
with latent space embeddings to discover interpretable latent
dynamics which can be simpler and easier to learn than their physical-space counterparts (Champion et al.
2019
Qian et al. 2020: Benner et al. 2020
Fries et al. 2022 He et al. 2023 Bonneville al. 2024a
Tran
et al.
2024
He al. 2025 Anderson et al. 2025).
Despite these advancements, many existing data-driven ROMs fail to enforce fundamental physical principles, such as energy conservation or the second law of thermodynamics, which are responsible for dynamical stability and long-term system behavior. As a result, their latent dynamics may fit given set of training data well while cxhibiting unphysical behavior under extrapolation or unseen conditions. To bridge this critical gap, thermodynamic priors have been introduced inlo dala-driven ROMs. Yu el al. Yu el al. (2021) introduced an OnsagerNet to learn stable and interpretable dynamics of dissipative systems by enforcing a generalized Onsager principle (Onsager] 1931). Geng et al. Geng ct al. 2025 extended Hamiltonian operator inference (Sharma et al. 2022; Gruber & Tezaur 2023 to port-Hamiltonian systems, inferring reduced operators from state/output snapshots and chosen Hamiltonian ansatz to yield ROMs that preserve energy storage, dissipation, and input output passivity. Park et al. Park et al. (2024) introduced a thermodynamics- informed latent space dynamics identification (tLaSDI) framework, which integrates nonlincar manifold Coupling (Grmela &
learning with GENERIC (General Equation for Non-Equilibrium Öttinger 1997 Ottinger 2005)) formalism-informeed neural networks (GFINNs) (Zhang et al. 2022) to learn latent dynamics that are consistent with thermodynaaical laws. In tLaSDI, the effective parameterization of latent dynamics is achieved through lyper--nntoencoder architecture, where hypernetwork (a et al., 2017) generates autoencoder coefficients based on input parameters. While this design enables flexible parameterization, it also introduces significant model complexity and training overhead, posing challenges for scalability in large-scale or high-dinensional parametric seltings.
To address this issue, this work proposes a novel and efficient tLaSD1 framework for parametric nonlinear dynamical systems. Our approach integrates standard antoencoders for nonlincar dimensionality reduction with newly developed parametric GFINNs (pGFINNs). Besides incorporating parametric dependence in simple and interpretable way, these networks are specifically designed to enforce the GENERIC structure, ensuring that the latent dynamics conserve free energy and generate entropy consistently across the param- eter space, thus satisfying both the first and second laws of thermodynamics. To further enhance model performance, we incorporate a physics-informet active learning strategy using a greedy, residual-based error indicator that adaptively selects the most informative parameter samples during training. This targeted sampling strategy enables effective generalization across parameter space, outperforming uniform sampling at the same computational cost.
The proposed tLaSDI framework is validated through numerical experiments on the Burgers' equation and the 1D/1V Vlasov-Poisson equation, which represent standard nonlinear benchmarks for model reduction and dynamics discovery. Compared to the hyper-autoennoder-beer parameterization, our method reduces
2

---

Published in Transactions on Machine Learning Research (02/2026)
training cost by 50%-90% and inference time by 57%-61%, enabling superior accuracy and interpretability at a greatly reduced computational cost. Furthermore, for the Vlasov-Poisson system, we achieve up to 2, 495x speed-up with less than 3% relative errors in a predictive setting, when compared to a traditional numerical simulation that produces a similar error with respect to the high-fidelity training data. Beyond these computational gains, our framework provides physically interpretable latent variables that reflect the system's 's thermodynamic behavior and offer insights into its evolution in both the latent and physical spaces. For example, the learned entropy in the Vlasov Poisson system is seen correspond to the growth rate in the two-stream instability, providing a physically relevant measure useful for further analysis.
In summary, this work represents a significant step toward the data-cfficient, interpretable, and physically- consistent reduced-order modeling of complex parametric systems, and positions the proposed LLLSDI as a robust foundation for next-generation ROMs that are both scientifically principled and computationally scalable. Our principal contributions are:
the development of PGFINNs to identify thernodynamically consistent latent dynamics enforcing free energy conservation and entropy generation across varying system parameters;
the proposal of an efficient tLaSD1 framework for reduced-order modeling through the integration of standard autoencoders with the developed PGFINNs;
the integration of an active learning strategy into the proposed tLaSDD framework for adapt.ive parameter sampling on- the-fly, leading to improved generalization capabilities across parameter space at equivalent cost to uniform sampling.
the production of interpretable and physically latent dynamics, achieving up to 2, 495 speed-up with 1-3% errors on parametric benchmarks, well as significant. reductions in training (50-90%) and inference (57-61%) cost.
The remainder of this work details these contributions and their consequences. Additional implementation details necessary for reproducing the results in the body can be found in Appendix.
2 Methodology
We consider parametric dynamical system characterized by system of ordinary differential equations (ODFs)
((:: f(, E [O, u(0: ()( E ER",
(1)
where u(t: μ)) represents the parametric, time-dependent solution to the dynamical system, T R4 is the final time, f is an appropriate field function that guarantees the existence of the solution, and ug is the initial state of u, parametrized by μ E 'D C RNN.. Eq. ① can be considered a semi-discretized system of partial differential equations (PDE), in which the numerical solution to Fq. 1 can be obtained using explicit or implicit time integration schemes. For example, with the implicit backward Euler time integrator, we compute an approximate solution by solving the nonlinear system of equations u, = un-1 + At f,, where u, R ul p.) the approximate state and fn = f(u,; lai 14) the discrete velocity field. The residual function then expressed as
r(u Up 11) = un un A f.
(2)
Computing the time integration corresponding to Eq. (2 is often computationally expensive, especially when the ambient dimension of the solution N,, large. In this work, we aim develop an accurate and efficient ROM framework based on thermodynamically consistent latent space dynamics identification, which preserves the thermodynamic character of the physics described by Eq. (1) in a machine-learned latent space where the governing dynamics have been distilled. The remainder of this section describes the proposed pGFINN architecture, its integration with tLaSDI, and the physics-informed active learning strategy which enables improved generalization performance.
2.1 Parametric GFINN (PGFINN)
The GENERIC formalism Grmela & Öttinger 1997 Öttinger 2005p, also known as the metriplectic for-
malism Morrison 1986 2009), mathematical framework describing beyond-equilibrium thermodynamic
3

---

Published in Transactions on Machine Learning Research (02/2026)
systems, including both conservative and dissipative systems. Its equations of motion compactly ex- pressed as
2 == L(z) 0E
as dz (z)
(z) + (z)
Oz
subject to [zz) [[))² ((z) M(z)"
([z))== = 0,
(3)
05 (iz) Oz
OF
(z) M (z)
(z)
Oz
where z E I2 denotes the state variable, E resp. S are scalar functions representing the total energy resp. entropy of the system, and L resp. M are matrix-valued functions representing the skow-symmetric Poisson resp. symmetric positive semi-definite friction matrices. The final set of conditions prescribing the kernels of L and M can be shown conservation (zz) = 0 and non-decreasing entropy Š(z) 2 (Gruber al. 2023a). Due to these general and favorable properties, the model reduction and dynamics identification of SEENERCC/metripleee
instantaneously guarantee the first two laws of thernodynamics, i.e., energy
systems is
active area of development, including strategies such (Gruber et al.
2023a Lee et al.] 2021 Zhang
ct al. 2022: Gruber et al., 2023b; 2025). Recently, Zhang et al. proposed GFINNs (Zhang et al. 2022
designed to exactly satisfy the degeneracy conditions in Eq. (3) through structure-infrmee architectures
for the governing data L, M, E, S. However, GFINNs do not natively acommodate differential equations with parametric variability in their governing operators. In this work, we introduce parametrization into GFINNs to enhance their ability to handle such dynamical systems, ensuring that the learned solutions remain accurate in these situations.
To minimize the requirements of this approach, we consider the most general setting where no prior knowledge on the system Eq. 1 is assumed. Therefore, all data defining the GENERIC surrogate are approximated with carefully constructed neural networks (NNs), i.e. ENN. SNN, LNN, and M The architecture of PGFINN is illustrated in Fig. [ where the function represents either SNN or ENN and the matrix field represents either LNN or NN- The most delicate part of this construction involves automatic satisfaction of the symmelry and degeneracy conditions in Eq. (3): the nelworks associated with the fields A are designed to ensure V,G ker A for the appropriate G. Inspired by ideas from matrix factorization, A is modeled by
A := Q0
(4)
where QG is matrix-valued function constructed based skew-symmetie matrix fields Sj, 1,..., K, so that the j-th row of Qo is defined as (S)V. This leads to the equality V& = 0 and therefore AV2G- = 0, i.e, the required degeneracy condition holds. Moreover, the matrix field B is skew-symmelric (Bz) if A represents L and symmetric positive semi-definite (BM) if A represents M, modeled by two triangular NNs,
= B and BM = TM
To capture systems with parametric dependence, the networks associated with G and B are parametrized by μ, allowing the proposed pGFINN to effectively capture parameter dependence while guaranteeing free energy conservation and entropy generation at each point of the parameter space. More detailed analysis on the effectiveness and performance of the proposed pGFINN is presented in Appendix A.4
Z
NN
G
V2G
QG
X
A
skew-sym. matrices
μ
NN
B
Figure 1: Architecture of the proposed PGFINN with trainable parameters highlighted in yellow.

---

Published in Transactions on Machine Learning Research (02/2026)
2.2 Thermodynamics-Inoon Latent Space Dynamics Identification (tLaSDI)
The tLaSDI presented in (Park al. 2024) consists of lyper-autoencoder for nonlinear dimensionality roduction combined with GFINN for learning thermodynamically consistent latent dynamics. The hyper- autoencoder in tLaSDI integrates standard autoencoder (Hinton & Salakhutdinov 2006) with a hyper- network (Ha et al. 2017), achieving parametric latent representation of the system state through the generation of autoencoder coefficients based on a given input parameter μ.. While this formulation does capture some form of parametric dependence in the governing equations, the hypernetwork- based parame- terizalion substantially increases network complexity and training overhead, which may limit the scalability of tLaSDI for large-scale parametric dynamical systems.
To improve the efficiency and performance of parametric tLaSDI, propose bypassing this hypernetwork entirely, instead integrating standard autoencoder with pGFINN, as described in Section 2.1 This has the advantage of incorporating parametric dependence directly into the learned dynamics, thereby maintain- ing consistency with the model from Eq. ( and enabling an easier learning problem for the dimensionality reduction. Fig. [ iillustrates a schematic of the proposed tLaSDI in inference mode, after both the autoen- coder and pGFINN have been trained. This inference proceeds as follows. Given a high-dimensional initial state Ug associated with the parameter μe, the encoder &e first transforms Ug a low-dimensional latent state 2o- The PGFINN #PGFINN then predicts thernodynamically consistent latent dynamics beginning zo. which are subsequently projected back to the physical space by the decoder &a.
Predicted Physical Dynamics
Initial State
uo
Φ e
Zo
4 PPGFINN
Z+
Φ d
ü+
μ
t
Time
Thermodynamically Consistent Latent Dynamics
Figure 2: Schematic of tLaSDI for parametric dynamical systems in inference mode.
The loss function used to train these constituent parts is sum of four terms:
(()) Limt (0) + Aree Croe + A Iac + Amor Emod (0):
(6)
where 0 is the collection of all trainable paramcters. The first loss term Line represents the integration error,
(un+1) - (u - (att
Lint ΣΣ J
a
where the integral approximated via time integration scheme, c.g., a. Runge- Kutta method. This term punishes discrepancy between the prredctt latent state Zn+1 = de (u,+1) and the corresponding one- step integration of the latent dynamics starting from the previous encoder-pon prrodee-predicted latent state 2,, (,,
5

---

Published in Transactions on Machine Learning Research (02/2026)
Said differently, the integration term enforces the approximate equality
[
1n11
(un+)))) ~ (un) +
pPFFNNN(Z()) (I,
(8)
In
which provides consistency condition between PGFINN integration and the dynamics encoder. The second loss term Lree represents the usual reconstruction error of the autoencoder, defined as
Lree ΣΣ da (Q.
(9)
which encourages the autoencoder to learn low-dimensional representations which effectively compress the solution data. The third loss term LJa represents an infinitesimal regularity condition call the Jacobian loss of the autoencoder,
Lac (I)((-J(u.))
(10)
Here, J(-) = Ja ()) represents the Jacobian of the autoencoder, i.e., the derivative of the decoder's output with respect to the encoder's input and, Ja(-) resp. Ja(+) represent the Jacobian of the decoder resp. encoder. This is heuristic, Sobolev-type loss that punishes infinitesinal inconsistency in the state u, and its reconstruction after autoencoding, enforcing the condition
d La ( (u.( (u,)ú, (un (u,)
ún
(11)
In the case where the derivative data ún is unavailable, the Jacobian loss can be alternatively defined
Lac ΣΣ 1- J(un) 2
I((n))
(12)
where I|- denotes the Frobenius norm. Finally, observe that the Jacobian of the autoencoder is rank- deficient since the latent dimension much smaller than the physical dimension, implying that J(.) # I is never equal to the identity mapping. This fact leads to the final loss term Lmod; which accounts for the modeling error in tLaSDI. In particular, an addition of zero followed by the Cauchy-Schward inequality leads to the statement.
( J(u,)) un s ú, Ja(o. wOFINN (u.(),)) + (u,))l (u, pPPFINN( (u.)
(13)
where the first term on the right-hand side quantífies the modeling error the full-order dynamics, obtained by projecting the latent dynamics predicted by pGFINN through the decoder. Conversely, the second term measures the modeling error between the encoder-derived representation of the velocity field, J, (u,)), and the PGFINN prediction, 2b (pCFIIN((.(U.))) (u.)). Accordingly, Lmod is defined as
Lmod ΣΣ ún - Ja(b ((,...))))),cIixxxx. |2 Jo(un) -t UPGFINN (u.))
(14)
The comprehensive loss function Eq. 6 reflects an error bound for tLaSDI under the assumptions that the decoder's Jacobian is bounded and Lipschitz continuous, and that the pGFINN predictions are bounded (Park et al. 2024). Advantageously, it enables simultaneous training of the autoencoder and the latent space dynamics model, ensuring consistency in the learned latent dynamics while minimizing modcling errors in the full-order system. The advantages of such joint training have been previously demonsttated in the literature (Champion et al. 2019 He et al. 2023 Bonneville et al.] 2024a Vijayarangan et al.) 2024 Park ct al. 2024 He et al.] (2025). highlighting that enforcing interactions between the autoencoder and dynamics models, and thereby imposing structure on the latent dynamics, is crucial for improving model performance and generalization. An upper bound the tLaSDI state error given in terms of the loss function Eq. (

---

Published in Transactions on Machine Learning Research (02/2026)
is presented in Appendix A.1 offering theoretical foundation for this training approach. More detailed discussion about the effects of the Jacobian and modeling loss terms model performance can be found in the Appendix A.2
Note that constructing the full tLaSDI loss in Eq. (6) requires derivative information as training data, especially for the Jacobian loss in Eq. (1) and the modeling loss Eq. [1).. In the case that this is prohibitive (such in the presence of noisy conditions), the strong form ODE in Eq. (I) can be transformed to weak form by multiplying both sides with continuous, compactly supported test function and inte- grating over its support. By applying integration by parts and exploiting the compact support of the test function, temporal derivatives of the state variables are transferred to the test function, thereby stabilizing derivative computation in the presence of noisy data and eliminating the need for pointwise derivatives of the input [Messenger & Bortz 2021a). Recent studies have successfully incorporated this weak formulation into latent-space dynamics identification frameworks to improve the robustness and accuracy of ROM in the presence of noise (Tran al. 2024 He al. 2025 Bonneville al.) 2024b). The proposod tLaSDI method be extended in similar manner, although this is beyond the scope of the present study.
2.3 Physics-Infone Informed Active Learning
Another important component of tLaSDI training relies on adaptive sampling. To effectively explore the parameter space and enhance model performance, tLaSD1 adopts greedy, physics-informed active learning strategy (He et al. 2023 which adaptively samples informative training data on the fly. Given testing parameter ,z,, modeling accuracy is assessed using physics-informed error indicator derived from the
residual Eq. (2) of the governing equations Eq.
Gres Σ
|r(ūn; ūn-1,
(15)
where ů represents approximate full-order dynamics through both the autoencoder and the latent dynamics learned by PGFINN. Typically, the sum on time steps n is sparsified, since predictions at a small subset of time steps suffice to obtain reasonable measurement of the error. This residual-based error indicator depends solely on model predictions and shows strong positive correlation with the maximun relative error, enabling efficient and effective adaptation across parameter space. When adaptive sampling is enabled, tLaSDI training begins with limited number of parameter points (training samples), often located at the corners of the parameter space. Every Nap epochs, the tLaSD1 model is evaluated at random set of testing parameters, computing the error indicator defined in Eq. (15) at each parameter value. The parameter point corresponding to the maximum error is then added to the training set before training resumes. This physics- informed active learning process continues until predefined number of training samples is reached or the desired model accuracy is achieved. Note that the mild computational overhead incurred by this procedure is generally outweighed by its performance improvement over simple uniform sampling (c.f. Section 3.1 leading to simple and robust boost to the generalization capabilities of the proposed tLaSDI model.
3 Experiments
We now demonstrate the performance of the proposod tLaSDDI framework by applying it common bench- marks in nonlinear model reduction: the 1D Burgers' equation and the 1D/1V Vlasov two-stream plasma instability problem.

---

Published in Transactions on Machine Learning Research (02/2026)
3.1 One-dimensiona Burgers' Equation
We begin by considering one-dimensional (ID) inviscid Burgers' equation with periodic boundary conditions and a Gaussian initial condition parameterized by {a, E D
du +
du = 0, E [0,
E [-33]
a1
Or
a(t, x- 3) =u(t, -3)
(16)
x²2
u(t = 0, x; 1e) = cxp
2w
The parameter space is defined as D (0.7,0.9 x [0.9, 1.1], associated with the magnitude and width of the initial Gaussian pulse. The implementation details can be found in Appendix
The hypernetwork-basee tLaSDI and the proposed PGFINN-based tLaSDI are trained with 9 parameter points uniformly distributed in the parameter space D. Fig. presents the maximum relative errors of the trained models across this parameter space, where Figs. 3 a- show that PGFINN-based tLaSD1 outperforms the hyper-autoencoder
based approach, highlighting the effectiveness of the proposed pGFINN parameterization. Moreover, the training cost of CGFINN-based tLaSDI is reduced by 50% over the previous hypernetwork variant, while the inference cost is lowered by 61%, demonstrating that the proposed approach also provides significant reductions in computational burden.
To demonstrate the effectiveness of the invsice-nnforme active learning strategy described in Section 2.3 we train an additional pGFINN-basod tLaSDI integrated with this strategy. The training begins with parameter points located at the corners of the parameter space and adaptively selects 4 additional parameter points using the active learning approach, resulting in total of 8 training points by the end of training. Although the total number of training points is less than that used in the case of uniform sampling, Figs, (b-c) demonstrate that the performance of PGFINN-based tLaSDI is further enhanced by the proposed active learning strategy, without increasing the cost of model training. Notably, active learning adaptively selects additional parameter points in regions of higher modeling error, particularly near the lower end of the parameter space, as shown in Fig. Bb). This leads to enhanced model accuracy across the entire range of parameters considered, significantly outperforming the case of uniform sampling especially in regions where errors are highest.
co
071
09
0603 14 05
077
07 OT
C17
05 03 21 CT
01
SE
073
03
04-00 22 27 07 OT
(1) 03 x
0505
24 04
oF
MC 22
15
021
06
0M
tm 3144 SE
08
0.0220
0907
35 03 09
CT
( 03 ML 07
09
21
EM 0257 27
03
1110
39
MAMAM wtt.h
Wdth
Mith
Witth
(a) hyper-antoencoder Uniform
(b) PGFINN Uniform
(c) PGFINN Adaptive
Figure 3: Comparison of the maximum relative errors across the parameter space from (a) hypernetwork- basod tLaSDI with uniform sampling, (b) pasdNN-based tLaSDI with uniform sampling, and (c) PGFINN- based (LASDI with adaptive sampling. The training points are highlighted with black boxes.
3.2 1D/1V Vlasov-Poisson Equation
The effectiveness of the proposed tLaSDI framework is further examined using the ID/IV Plisov-Poisson equation with initial conditions parameterized by the temperature T' and the wavenumber k; i.e., /
8

---

Published in Transactions on Machine Learning Research (02/2026)
(T E D= [0.9, 1.1|
[100, 1.2].
These equations given by
0f
Ouf
0
dΦ
0,
t [0,
I e 2T]
v [-7,7]
01
dr
Ov
dr
dΦ
fdv
(17)
dx²
(u - 2)² 2T
(-(u+ 2)2 ]
f(t O, O, 21
1
cos (kz)
exp
+ exp
2
10
2T
where f(x, u) is the plasma distribution function, dependent spatial coordinate and velocity ordinate v, and + is the electrostatic potential. This model describes the dynamics of one-dimensional, collisionless, electrostatic plasma, and is representative of the complex models for plasma behaviors observed in nuclear fusion reactors. Owing to dependence on the velocity variable, a two-dimensional PDE despite single spatial dimension. The physical dynamics corresponding to the parameter case (T = 0.9, 1.0)
are illustrated in Fig. 4
step:
step: 16
step: 33
step: 50
step: 66
step: 83
step: 100
step: 116
step: 133
step: 150
step: 166
step: 183
step: 200
step: 216
step: 233
step: 250
Figure 4: 1D/1V Vlasov The physical dynamics of the parameter case (7 0.9, k 1.0).
In this more challenging example, the training data include 16 parameter points, uniformly distributed across the parameter space D. The implementation details can be found in Appendix A.5 Figs. aa-b) compare the latent dynamics learned by hypernetwork-busee (LaSDI and the proposed pGFINN-based LLaSDI. Observe that the proposed tLaSDI model learns simpler latent dynamics and exhibits stronger consistency between the latent state encoding and the PGFINN dynamics prediction. This is further confirmed by examining the frequency-domain representation of the latent dynamics, shown in Figs. 5 B(cdd,, which reveals that the frequency content of the latent dynamics generated by the proposed tLaSDI concentrate at lower Hertz values than those in the model with hyper-autoencoder, further highlighting the simplicity of the learned dynamics.
1.0
Hy pen--mmendee
1.000
Lacuder PUPINN
2.0
CP INN
065
Nomeatzee hnpppt..c.
0.2
Nomattors
0.00
n0
40.50
-0.6
-0.5
20
650
025
673
1.0
350
3.26 200
50
040
025
039
130
180 3.76 200
Time
Time
(a) hnpper-uuoenccoder
(b) PGFINN
(c) hyper-autoencoder
(d) pGFINN
Figure 5: 1D/1V Vlasov Comparison of the latent dynamics learned by (a) hypernetwork-based tLaSDI and (b) pGFINN-based tLaSDI; the corresponding frequencies for (c) hypernetwork- based tLaSDI and (d) PGFINN-based tLaSDI.
Figs. demonstrate that the simpler and more consistent latent dynamics of the proposed tLaSDI also contribute to higher model accuracy, achieving average error of 1.66% across the entire parameter space,

---

Published in Transactions on Machine Learning Research (02/2026)
compared to 2.18% for its hyper-autoencoded counterpart, 24% reduction in error with remarkably less training complexity. The significantly lower errors observed at the training parameter points for the proposed tLaSDI suggest that the pGFINN which propagates latent dynamics more effectively captures parameteriza- tion in the governing Eq. ( as opposed to the hyper-antocncodeed parameter-dependenoe alongside static latent dynamics. Additionally, due to the comparatively smaller size of the pGFINN-based ILaSDI, model accuracy is expected to improve as the training dataset expands with less risk of overfitting Furthermore, the training cost of the proposed tLaSDI is reduced by 90%, while the inference cost is lowered by 57% indicating highly impactful reduction in computational burden. Compared with the high-fidelity simula- tion code (HyPar (Ghosh] 20255) running a spatially coarsened configuration that produces similar error (1.66%) with respect to the training data, the proposed tLaSDI model achieves a 2, 494 speed-up. The time step size used to generate the training data is 5 x 10-3, Increasing the time step size of the high-fidelity simulations to 2 x 10 "hile keeping the spatial resolution fixed reduces the computational time and incurs small relative errors of 0.014% with respect to the training data computed at the smaller time step size. Further increasing the time step size leads to unstable simulations and convergence issues. When compared to the high-fidelity, simulation with the largest time step size of 2 10-?, the proposed tLaSDI model achieves 1,370x speed-up. Although the examples in this paper rely on simulated data generated from PDE solvers, we emphasize that the proposed tLaSDI model is equally applicable to settings involving dynamic sensor data where the governing ODEs/PDEs are not. known a priori. More detailed discussion about the speed-up analysis can be found in the Appendix A.3
0.90
20
1
ER
18
17
10
0.1
1/
1
1.8
20
0.91
20
14
1.4
1.4
1:4
1M
1.5
3.0%
3.0%
082
24
22
20
10
17
17
17
17
18
10
19
18
18
19
19
18
19
19
18
082
31
25
24
24
22
20
18
18
1.5
17
18
18
18
19
29
20
20
1.8
083
25
23
21
1.9
1.8
1.8
18
1.8
1.8
1.9
20
20
20
20
20
20
20
20
20
21
0.83
35
2.
255
26
24
22
1.0
1.9
1.8
1.7
1.7
1.8
1.8
1.0
200
20
2.0
21
2.1
2:1211
2.0
094
20
19
19
19
21
21
21
21
21
0.94
2.3
20
18
18
18
19
21
211
20
19
085
21
20
20
20
22
22
21
3.0%
0.85
1,5
1.5
1.G
1.0
3.0%
087
21
27
20
21
29
27
7
37
27
21
12
13
144
1.6
16
1.
088
28
26
24
23
22
21
20
20
20
21
21
211
22
22
22
22
22
22
211
21
2.0
0.88
1.8
1.6
1.4
1.4
14
1.4
1.
1.3
1.4
1.4
1.4
144
1.4
1:4 1.5
1.6
1.
1.5
1.6
1.6
0.9
2
24
23
21
21
20
20
26
21
21
21
21
21
22
22
21
21
21
20
18
2.55%
0.90
210
1.6
1.5
16
1
15
16
17
1.
1.
1V
1.M.
16
1.S
15
1.S
1.
17
1.
2.55%
1.00
2.8
23
22
21
20
20
20
20
20
21
21
21
211
21
21
21
20
1.9
18
1-
1.00
20
1.9
20
20
20
1.0
1.9
1.8
2.
1.9
1.9
1.0
1.7
1.7
1.8
1.7
1011
25
2
22
21
20
20
19
20
21
21
21
21
20
20
19
19
18
0.01
1.8
1.8
17
1.
16
1:02
2.0
20
19
1.
22
22
22
2.0%
1.02
14
1:03
10
22
1.03
1.5
1:04
25
21
2.
20
1.04
16
15
13
1.05
2.0
1.9
1.6
18
18
19
20
20
21
22
23
24
24
24
24
23
23
22
22
21
1.05
1.8
1.8
1.8
1.5
17
1.6
1.6
17
17
16
1.6
16
1.6
1:4
1.3
1.
13
1.3
14
1.5
1G
108
21
20
19
1
19
20
20
21
22
23
74
25
75
76
28
75
75
74
24
23
1.5%
1.0
20
22
21
20
18
17
TH
12
18
19
LS
17
TK
14
14
13
144
1.
15
1.696
1.07
25
24
24
24
2
2
26
28
28
26
26
24
1.07
2.1
22
22
20
1.9
17
17
17
18
1.9
1.9
1.8
1.5
1:4
14
1.4
1.5
1.6
108
25
30
1
37
34
37
37
31
1.08
20
19
18
16
1
18
2.0
699 110%.0
69 .9
6
07
(a) hyper-autoencoder
(b) PCFINN
Figure 6: ID/IV Vlasov Comparison of the maximum relative errors across the parameter space between (a) hypernedworkk-bssed (LSSDI and (b) PGFINN-basod (LaSDI. The training points are highlighted with black boxes.
To confirm the crincture-presevvatin properties of the proposed approach which are guaranteed by the PGFINN component, Fig. illustrates a roll out of the learned energy and entropy in the latent space and their rates for various parameter instances. As expected, the energy conservation and increasing entropy production, demonstrated in Fig. 7 confirm that the learned latent dynamics satisfy the first two laws of thermodynamics guaranteed by the GENERIC formalism. Moreover, these dynamics are remarkably interpretable as a consequence of GENERIC structurr-preeecration Observe from Fig. d( that, for fixed temperature T, increasing the wavenumber & leads to higher initial entropy and lower final entropy. Similarly, Fig. ( shows that, for fixed wavenumber k, increasing temperature T leads to a higher initial entropy and lower final entropy. These trends are consistent with known plasma physics in the high-dimensional space, since increasing temperature and wavenumber suppresses growth in the two-stream instability (Koide et al., 2023). This demonstrates the ability of the proposed LLaSDD to capture physically meaningful trends in the
10

---

Published in Transactions on Machine Learning Research (02/2026)
high-fidelltn system independently of its accuracy, since each realization is guaranteed to exhibit meaningful thermodynamics. Additionally, the entropy production rates shown in Fig. | ()) exhibit peaks at around steps 15, 50, 75, 120, and 180, which correspond meaningfully to the growth rate of the two stream instability. These results suggest that the thermodynamic behavior within the latent space may provide valuable insights into the dynamics occurring the physical space.
1n-7
1.1
km1.00
1.275
1m0.9C
1 1m0.90
k-1.05
T-0.95
T-0.95
1.20
0.9
k-1.
1.225
T=1.05
7 1-005
T=1.10
T-1.10
0.7
1.175
0.6
1.150
100
150
200
250
50
150
250
50
150
150
200
250
Stops
Staps
(a)
(b)
(c)
T=0.90
k-1.00
km.100
10100
1-0
05
45
c.1.0
T=100
1-00
T=105
1.00
e-120
T-1.0
1.30
-6
30
130
150
200
250
50
100
150
200
250
50
100
150
200
250
Staps
Staps
Stops
(d)
(e)
(f)
Figure 7: ID/IV Vlasov The evolution of energy for: (a) T 0.9 and varying k; (b) k 1.0 and varying
T; the evolution of energy rates for (c) & = 1.0 and varying T; the evolution of entropy for: (c) T 0.9 and
varying k; (d) k = 1.0 and varying T; the evolution of entropy production rates for (e) k 1.0 and varying T.
3.3 Baseline Comparison
In this section, we benchmark the proposed tLaSDI framework against two state-of-the-at operator learning methods, including Deep Operator Networks (DeepONets) (Lu et al.) 2021 2022) and the Fourier Neural Operator (FNO) Li ct al. 2020 The test problem considered the 1D/IV Vlasov-Poisson equation (sce Section 3.2).
For this experiment, a three-dimensions FNO (FNO-3D) is used that directly performs convolution in space and time. Both DeepONet and FNO-3D trained using the same dataset, described in Section 3.2 with implementation details provided in Appendix A.5) All models take the full-order initial state input and predict the solution at future time steps. To ensure a fair comparison, each model is configured to maintain a comparable architecture and number of trainable parameters to our proposed PGFINN-based tLaSDI (c.f. Appendix A.5.2).
Fig. shows the maximum relative errors across the parameter space for DeepONet and FNO-3D, which achieve average errors of 2.87% and 2.20%, respectively. Notably, DeepONet exhibits larger errors most regions of the parameter space, whereas FNO-3D demonstrates lower errors at training points compared to nearby testing points. This pattern is also observed in the proposed tLaSDI, shown in Fig. 6 (bb. Table compares the performance of the tLaSDI models against DeepONet and FNO-3D models. Remarkably, the proposed LLaSDI achieves superior accuracy across the parameter space in this example, with an average error of 1.66%.
While accuracy is certainly useful metric by which gauge model performance, it is important note additional advantages of tLaSDI which are unique to its construction. First, it be seen that both
11

---

Published in Transactions on Machine Learning Research (02/2026)
Table Comparison of performance on ID/IV Vlasov-Poisson equation
Model
Parameters
Time (s) Epoch
Mean Error (%)
LLSDDI (pGFINN)
480,367
0.053
1.663
tLaSDI (Hyper-Autoencoder))
9,526,427
0.527
2.176
DeepONet FNO-3D
481,361
0.042
2.870
486,336
0.687
2.200
DeepONet and FNO require parameterization through the initial condition, i.e,, they learn a direct mapping from the initial state to future states after (implicit) temporal evolution. However, in applications where parameters affect aspects of the dynamics other than the initial condition, such as the system properties a, m.k seen in previous experiments, the full-order initial state remains identical across the entire paramcter space. In such scenarios, DeepONet. and FNO cannot directly account for parameter dependence, thereby limiting both their performance and applicability. In contrast, the proposed parametrie tLaSD1 can naturally accommodate arbitrary forms of paramcterization by decoupling paramcter dependence from the initial state. Furthermore, DeepONel and FNO are presently limited to pre-defined sampling strategies, while LLaSDI leverages physics-infrrmed active learning for adaptive sampling, enhancing both its efficiency and predictive performance in generalization tasks.
Perhaps most importantly, unlike the black-box, nature of DeepONet and FNO, the proposed framework strongly enforces thermodynamics principles to enable interpretable, structure-preserving learning. This euforcement not only guarantees important mathematical properties of the learned dynamics such as global asymptotic stability, but also enhances generalization robustness and provides deeper insights into the un- derlying physical dynamics, without compromising predictive accuracy, as demonstrated in Section 3.2 The benchmark results seen here highlight the potential of tLaSDI as an efficient, interpretable, and physically consistent reduced-order modeling approach for parametric dynamical systems.
0.90
25
0.91
28
16
34
0.61
24
33
23
25
3.5%
-355
0.82
16
14
33
32
32
31
10
30
3.
10
10
30
30
31
31
31
32
33
0.52
19
1.5
22
26
20
27
22
15
23
80
35
32
24
18
26
27
25
22
1.9
0.80
35
33
3.2
3.1
3.2
31
30
2.3
2.0
25
23
2.
30
30
3.1
0.53
2.1
24
2.3
2.8
2.0
27
24
90
34
32
25
20
25
26
28
0.94
35
31
28
27
-3.0%
0.56
22
-30%
0.90
39
0.56
8
23
0.97
35
3.1
2.7
28
2
255
28
25
25
29
2.3
30
0.77
14
15
2:
1.8
97
15
27
33
84
2C
13
32
24
22
18
0.90
32
30
2m
2
2.6
26
255
2.5
255
25
25
25
2
2
20
2
24
2
0.50
16
1.5
21
21
24
14
1:5
20
3.4
24
14
22
23
22
1.8
15
0.89
35
3.2
3.0
27
2.7
2.7
27
27
20
2.
27
28
2.9
-7.5%
0.8
18
15
211
22
2:1
28
28
35
36
23
24
23
15
1.00
3.9
27
1.02
1.02
34
23
15
-2.0%
1.50
30
26
20
2.
25
25
20
75
25
28
2.7
1.03
14
16
22
22
18
12
18
34
28
1
12
20
22
22
18
1.24
40
2.5
3
28
28
26
2.5
2.5
2.5
25
25
25
5.5
25
25
2.
255
2.7
1:64
15
17
22
22
18
14
1:
28
34
29
20
14
24
1.00
3.1
3.0
26
26
25
2.
25
25
25
255
2.
1.05
24
18
23
1
24
1.00
2.0
211
21
25
22
22
25
1.00
35
36
2.
1.08
22
23
25
24
23
22
27
29
23
20
25
26
25
22
1.09
04
3.
3.
2
28
20
28
2.
26
26
26
2.0
2.
27
27
28
30
3.0
1.09
20
2.1
24
25
24
23
1.9
17
24
3.0
28
22
10
23
25
24
1.10
36
3.3
32
3.1
29
2.5
2.5
2.5
30
32
35
35
1.10
10
24
25
21
16
28
33
80
22
25
25
22
1.
10,00
110
(a) DeepONet
(b) FNO-3D
Figure 8: ID/IV Vlasov Maximum relative errors across the parameter space: (a) DeepONet (b) FNO-3D. The training points are highlighted with black boxes.
12

---

Published in Transactions on Machine Learning Research (02/2026)
Limitations
This work has restricted invostigation to problems with moderate parameter dimensionality and physical- state dimension. Although the proposed approach can, in principle, be extended to high-dimensional pa- rameter spaces and large-scale physical states, doing so would primarily serve to assess scalability and computational cost rather than the core methodological contributions. Accordingly, the experiments pre- sented here should be regarded as proof-of-concept benchmarks that validate the fundamental properties of the DGFINN+tLaSII framework, leaving a systematic study of scalability to future work.
4 Conclusion
We have presented enhanced, wowooddmmmmmiiiioononi latent space dynamics identification (tLaSDI) framework for efficient, interpretable, and physically consistent reduced-order modeling of parametric dynam- ical systems. The newly developed parametric GENERIC cermalism-informae neural networks (pGFINNs), designed for use within the proposed tLaSDI, guarantecs consorvation of free energy and generation of en- tropy in the latent space across variations in system parameters, ensuring thernodynamically consistent latent dynamics for any parametric realization. In addition, the integration of a physics-informed active learning strategy further improves model performance by adaptively selecting informative parameter sam- ples based a residual-based error indicator, leading to tLaSDI method exhibiting simple latent dynamics along with highly substantial improvements over previous methods on standard benchmarks. In particular, the proposed approach achieves a 50%-90% reduction in training cost and 57%-61% reduction in inference time over tLaSDI models incorporating paramotric dependence through a hyper-autoencoder. Compared to the high-fidelity simulation, it also achieves up to a 2, 495x speed-up with 1-3% relative errors. Perhaps most importantly, the learned latent dynamics reveal interpretable thermodynamic behavior, offering valuable in- sights into the physical system's evolution which can be used to guide control and analysis. Ultimately, these results highlight the potential of the proposed tLaSDI framework scalable, data-efficient, and physi- cally grounded foundation for the next generation of reduced-order models in complex, high-dimensional parametric settings.
Although the experiments in this study focus on parameterization through initial conditions, the proposed framework is readily extensible to other forms of parameterization, such as material properties and constitu- tive relationships, which are particularly relevant for inverse problems. While the use of a physics-informed error indicator for active learning relies on access to the high-fidelity solver, which may not be feasible in data-only scenarios or certain practical applications, the framework can be adapted by incorporating a Gaus- sian procoss-based active learning strategy for adaptive sampling [Bonneville et al.] (024a). Since derivative computations may become oscillatory in scenarios involving noisy data, thereby affecting latent dynamics identification and model performance, a weak-form formulation can be leveraged to reduce variance and achieve robust latent space dynamics identification under noisy conditions (Messenger Bortz 2021a Tran ct al. 2024 He al.] 2025 Bonneville et al.] 2024b). Moreover, the proposed tLaSDI framework is general and not limited to the use of standard autoencoders; alternative dimensionality reduction techniques, such convolutional autoencoders or linear compression techniques like POD, may be employed to improve training efficiency. Future work will consider the integration of an automatic neural architecture search (Pham et al., 2018 Ren et al. 2021) to further optimize the autoencoder design for enhanced generalization performance.
Acknowledgments
This work was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Advanced Scientific Computing Research (ASCR), through the CHaRMNET Mathematical Multifaceted Integrated Capability Center (MMICC) under Award Number DE-SC002316 to YC, and through the SEA-CROGS MMICC under Award Number DE-SC0023191 supporting AG. Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration under Contract
LLNL document release number: LLNL-CONF- 2006917. This article has been co-authored by an employee of National Technology Engincering Solutions of Sandia, LLC under Contract No. DE-N A0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title and interest in and to the article and is solely responsible for its contents. The United
13

---

Published in Transactions on Machine Learning Research (02/2026)
States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of foderally sponsored rescarch in accordance This
with the DOE Public Access Plan https://www.ent wwergy 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. KL acknowledges support from the U.S. National Science Foundation under grant ITS 2338909. YS was partially supported by the National Science Foundation under the grant DMS- 2513966 (Computational Mathematics Program) and the NRF grant funded by the Ministry of Science and ICT of Korea, (RS-2023-002119989)
References
Moloud Abdar, Farhad Pourpanah, Sadiq Hussain, Dana Rezazadegan, Li Liu, Mohammad Ghavamzadeh, Paul Fieguth, Xiaochun Cao, Abbas Khosravi, U Rajendra Acharya, Vladimir Makarenkov, and Saeid Na- havandi. A review of uncortainty quantification in deep learning: Tochniques, applications and challenges. Information fusion, 76:243 297, 2021.
William Anderson, Scung Whan Chung, and Youngsoo Choi. mlasdi: Multi-stage latent space dynamics identification. arXiv preprint orXiv:2506.0920 2025.
Peter Benner, Pawan Goyal, Boris Kramer, Benjamin Peherstorfer, and Karen Willcox. Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms. Computer Methods in Applied Mechanics and Engineering, 372:113433, 2020.
Gal Berkooz, Philip Holmos, and John L Lumley. The proper orthogonal dccomposition in the analysis of turbulent flows. Annual review of fluid mechanics, 25(1):539-575, 1993.
Anders Bondeson, Thomas Rylander, and Pär Ingelström. Computational electromagnetics. Springer, 2012.
Christophe Bonneville, Youngsoo Choi, Debojyoti Ghosh, and Jonathan L Belof. GPLaSDI: Gaussian process-based interpretable latent space dynamics identification through deep autoencoder. Computer Methods in Applied Mechanics and Engineering, 418:1 16s11656 6535, 2024a.
Christophe Bonneville, Xiaolong He, April Tran, Jun Sur Park, William Fries, Daniel A Messenger, Siu Wun Cheung, Yeonjong Shin, David M Bortz, Debojyoti Ghosh, al. A comprehensive review of latent space dynamics identification algorithms for intrusive and non-intrusive reduecd-ordder-modeling arXiv preprint arXiv:2403. 1004440888 2024b.
Steven Brunton, Joshua Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 113 (15):3932 3937, 2016.
Kathleen Champion, Bethany Lusch, J Nathan Kutz, and Steven I. Brunton. Data-driven discovery of coordinates and governing equations. Proceedings of the National Academy of Sciences, 116(45):22445- 22451, 2019.
William D Fries, Xiaolong He, and Youngsoo Choi. LaSDI: Parametric latent space dynamics identification. Computer Methods in Applied Mechanics and Engineering, 399:115436,, 2022.
Lawson Fulton, Vismay Modi, David Duvenaud, David I. W. Levin, and Alec Jacobson. Latent-space dynamics for reduced deformable simulation. Computer Graphics Forum, 38(2):379-38) 391, 2019. doi: 10. 1111/cgf.1366
David Galbally, Krzysztof Fidkowski, Karen Willcox, and Omar Ghattas. Non-linear model reduction for uncertainty quantification in large-scale inverse problems. International journal for numerical methods in engineering, (12) 2010.
14

---

Published in Transactions on Machine Learning Research (02/2026)
Yuwei Geng, Lili Ju, Boris Kramer, and Zhu Wang. Data-driven reduced-order models for port-hamiltonian systems with operator inference. Computer Methods in Applied Mechanics and Engineering, 442:1 18042, 2025.
Debojyoti Ghosh. Hypar repository. httge///atbackee.orr///aggg////p 2025. Software.
Miroslav Grmela and Hans Christian Öttinger. Dynamics and thermodynamics of complex fluids. i. devel- opment of a general formalism. Physical Review E, 56(6))6620, 1997.
Anthony Gruber and Irina Tezaur. Canonical and noncanonical Hamiltonian operator inforence. Computer Methods in Applied Mechanics and Engineering, 416:116334, 2023.
Anthony Gruber, Max Gunzburger, Lili Ju, and Zhu Wang. Energetically consistent model reduction for metriplectic systems. Computer Methods in Applied Mechanics and Engineering, 404:115709, 2023a. ISSN 0045-7825. doi: https://ddi.orrg/100. 1wt.//0......///.2.. 2mw.2.02.1117000000 URL
Anthony Gruber, Kookjin Lee, and Nathaniel Trask. Reversible and irreversible bracket-based dynamics for deep graph neural networks. In Thirth-seventh Conference on Neural Information Processing Systems, 2023b. URL ociecc//Arrteeee//////095522226
Anthony Gruber, Kookjin Lee, Haksoo Lim, Noscong Park, and Nathanicl Trask. Efficiently parameterized neural metriplectic systems. In The Thirleenth International Conference on Learning Represenlations, 2025. URL htttps://openreviieee wet./oorumiddullliaBaa
David Ha, Andrew M. Dai, and Quoe V. Te. Hypernetworks. In International Conference on Learning Representations, 2017. URL htttp:///pppaaoooeee...e////rrrs
Xiaolong He, Youngsoo Choi, William D Fries, Jonathan L Belof, and Jiun-Shyan Chen. gLaSDI: Parametric alysics-informed greedy latent space dynamics identification. Journal of Computational Physics, 489: 112267, 2023.
Xiaolong He, April Tran, David M Bortz, and Youngsoo Choi. Physics-informed active learning with simul- taneous weak-form latent space dynamics identification. International Journal for Numerical Methods in Engineering, 126(1):c7633) 2025.
Geoffrey E Hinton and Ruslan R Salakhutdinov. Reducing the dimensionality of data with neural networks. science, 313(65))):504 550-5077 2006.
Guang-Shan Jiang and Chi-Wang Shu. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126(1):202-222)) 228 1996. ISSN 0021-9991. doi: 10.1006/jcph.1900 http://jph...99.00110 1996.0130.
David Jones, Chris Snider, Aydin Nasschi, Jason Yon, and Ben Hicks. Characterising the digital twin: A systematic literature review. CIRP journal of manufacturing science and technology. 29:36 52, 2020.
Youngkyu Kim, Youngsoo Choi, David Widemann, and Tarck Zohdi. A fast and accurate physics-informed neural network reduced order model with shallow masked antoencoder. Journal of Computational Physics, 451:110841, 2022.
Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXio preprint 1rXiv:1412.60 6980, 2014.
Shinji Koide, Masaaki Takahashi, and Rohta. Takahashi. A relativistic two-stream instability in extremely low-density plasma. Physics of Plasmas, 30(6), 2023.
June Gunn Lee. Computational materials science: an introduction. CRC press, 2016.
Kookjin Lee and Kevin T Carlberg. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. Journal of Computational Physics, 404:108973, 2020.
15

---

Published in Transactions on Machine Learning Research (02/2026)
Kookjin Lee and Eric J Parish. Parameterized neural ordinary differential equations: Applications to com- putational physics problems. Proceedings of the Royal Sociely A, 477 (22(22533:202211662 2021.
Kookjin Lee, Nathaniel Trask, and Panos Stinis. Machine learning structure preserving brackets for fore- casting irreversible processes. Tn M. Ranzato, A. Beygekzmee, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan (eds.), Advances in Neural Information Processing Systems, volume 34, pp. 5696 5707. Cur- ran Associates, Inc., 2021. URL /tstpp::/proceeeee wwweeediinnnnn.uuiss..o. http:///ppppriieeee///////////// 22lccd2276600336228886666----222
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint Arx::20010. 08895, 2020.
Mengnan Liu, Shuiliang Fang, Huiyue Dong, and Cunzhi Xu. Review of digital twin about concepts, tech- nologies, and industrial applications. Journal of manafacturing systems, 58:346 361, 2021.
Lu Lu, Pengzhan Jin, Guofei Pang. Zhongqiang Zhang, and George Em Karniadakis. Learning nonlincar operators via DeepONet based on the universal approximation theorem of operators. Nal. Mach. Intell., 229, 2021.
Lu Lu, Xuhui Meng, Shengze Cai, Zhiping Mao, Somdatta Goswamii, Zhongqiang Zhang, and George Em Karniadakis. A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data. Computer Methods in Applied Mechanics and Engineering, 393:114778, 2022.
Romit Maulik, Bethany Lusch, and Prasanna Balaprakash. Reduced-order modeling of dddvection-dominate systems with recurrent neural networks and convolutional autoencoders. Physies of Fluids, 33), 2021.
Sean McBane and Youngsoo Choi. Component-wise reduced order model lattice-type structure design. Computer melhods in applied mechanics and engineering, 381:113813, 2021.
Daniel A Messenger and David M Bortz. Weak SINDy for partial differential equations. Journal of Compu-
lational Physics, 143:110525, 2021a.
Daniel A Messenger and David M Bortz. Weak SINDy: Galerkin-based data-driven model selection. Multi- scale Modeling & Simmlatiin, 19(3):1474-1497, 2021b.
Philip J. Morrison. paradigm for joined hamiltonian and dissipative systems. Physica D: Nonlineur
Phenomena, 18(1):410 419, 1986. ISSN 0167-2789. doi: http:://looooog///110/////0 (2200/006677777790888222222222 URL http:///ww...oocccccccccccccctts
Philip J. Morrison. Thoughts on brackets and dissipation: old and new. Journal of Physics: Conference
Series, 11000)), 2009. doi: 10.1088 1726-//4/66119///////1
William L Oberkampf and Timothy G Trucano. Verification and validation in computational fluid dynamics. Progress in aerospace sciences, 38(3):209 272, 2002.
Lars Onsager. Reciprocal relations in irreversible processes. i. Physical review, 37(4):405, 1931.
Jun Sur Richard Park, Siu Wun Cheung, Youngsoo Choi, and Yeonjong Shin. tLaSDI: Thermodynamics- informed latent space dynamics identification. Computer Methods in Applied Mechanics and Engineering, 429:117144, 2024.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-performance deep learning library. Advances in newral information processing systems, 32, 2019.
Anthony T Patera and Gianluigi Rozza. Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. MIT Cambridge, 2007.
16

---

Published in Transactions on Machine Learning Research (02/2026)
Benjamin Peherstorfer and Karen Willcox. Data-driven operator inference for nonintrusive projection-based model reduction. Computer Methods in Applied Mechanics and Engineering, 196:196-2115 2016.
Hieu Pham, Melody Guan, Barret Zoph, Quoe Le, and Jeff Dean. Efficient neural architecture search via parameters sharing. In International conference on machine learning, pp. 4095 4104. PMLR, 2018.
Elizabeth Qian, Boris Kramer, Benjamin Peherstorfer, and Karen Willcox. Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems. Physica D: Nonlinear Phenomena, 406: 132401, 2020.
Pengzhen Ren, Yun Xiao, Xiaojun Chang, Po-Yao Huang, Zhihui Li, Xiaojiang Chen, and Xin Wang. comprehensive survey of neural architecture search: Challenges and solutions. ACM Computing Surveys (CSUR), 54(4):1 34, 2021.
Michael Schmidt and Hod Lipson. Distilling free-form natural laws from experimental data. science, 324 (5923):81-85) 2009.
Daniel V Schroeder. An introduction to thermal physics. Oxford University Press, 2020.
Harsh Sharma, Zhu Wang, and Boris Kramer. Hamiltonian operator inference: of reduced-order models for canonical hamiltonian systems. Physica Nonlineor Phenomena, 431:133122, 2022.
learning
Ole Sigmund and Kurt Maute. Topology optimization approaches: A comparative review. Structural and multidisciplinary optimization, 48(6):1031-1053 2013.
Jos Thijssen. Computational physics. Cambridge university press, 2007.
April Tran, Xiaolong He, Daniel A Messenger, Youngsoo Choi, and David M Bortz. Weak-form latent space dynamics identification. Computer Methods in Applied Mechanics and Engineering, 427:116998,, 2024.
Vijayamanikandan Vijayarangan, Harshavardhana A Uranakara, Shivam Barwey, Riccardo Malpica Galassi, Mohammad Rafi Malik, Mauro Valorani, Venkat Raman, and Hong G Im. A data-driven reduced-order model for stiff chemical kinetics using dynamics- informed training. Energy and AI, 15:100325, 2024.
Daniel A White, Youngsoo Choi, and Jun Kudo. A dual mesh method with adaptivity for stress- cresss-oontt topology optimization. Structural and Multidisciplinary Optimization, (1 2020.
Haijun Yu, Xinyuan Tian, Weinan E, and Qianxiao Li. OnsagerNet: Learning stable and interpretable dynamics using generalized onsager principle. Physical Review Fluids, 6(11):114402, 2021.
Zhen Zhang, Yeonjong Shin, and George Em Karniadakis. GFINNs: GENERIC formalism informed neural networks for deterministic and stochastic dynamical systems. Philosophical Transactions of the Royal Society A, (8002229)2202102000 2022.
Hans Christian Öttinger. Beyond Equilibrium Thermodynamics. John Wiley & Sons, 2005.
A
Appendix
A.1 Error Analysis
This section provides a bound on the error of the tLaSDI approximation similar Park et al. (2024),, along with proof for completeness. Note that parametric dependence will be largely ignored in the following arguments; the resulting bound be assumed to hold pointwise for cach value of the parameters μ. First, observe that the error in the LLaSDI approximation is expressible
e(!) u(t) i(() (u(() a()) (a(() -(())
(18)
17

---

Published in Transactions on Machine Learning Research (02/2026)
where ü(t) (u(o)(u(t)) represents the reconstructed full-order dynamics from the autoencoder and ū(t) Φ (z())) denotes the tLaSDI approximation. Assuming the initial latent state z(to) &. (u(to)), the latent tLaSDI dynamics are given by á f" (2) where f" denotes the PGFINN. The goal is to bound the method error e(I) in lerms of quantities related to the loss terms Lint; Lre,, SJac, and Lmnd-
To this end, observe that eAB(t) u(t) - â(t) accounts for the reconstruction error of the autoencoder, whereas (ti = â(t) - ū(t) accounts for the error of latent space dynamics identification. It follows that there is an initial value problem for the reconstruction error,
ĒAE E --= ú.
eAE(to) -((o)) Da )a(
(19)
Integrating this time produces the solution
EAE eee((t) + S (I- ((((s())
(20)
which can be bounded for any [0,
[|)) s a (lese(to)ll [ |[(T- J(()) + (1 -a)llexe(()--
(21)
Turning attention to the latent dynamics identification, there is corresponding initial value problem,
épí = á - ú= J(u)f(u)
- j( (z), ena(ta) =
(22)
Observe that an addition of zero on the right-hand side leads to
(u))f(u) - (a(2) (z) (J) (a(o. (z)) ( Ja(z)
(23)
which can be bounded with the triangle inequality to yield
[[())(())))) - Ja(z) s (w))ll (1) f'(2]]
(24)
+ {IJa( - Ja(z)] ||
On the other hand, a different addition of zero gives the alternative expression
(())((u) - Ja(z) = (J())(()) -(()) + (f(u) (z)
(25)
which is similarly bounded as
J[u)f(u)) - J. Ja(z) ) [f(()) (z(z)
(26)
Therefore, o-linear combination of Eqs.
24
and 26
yields
[u)){(( - Ja a Ja (w))l (d.(()((u)
(u) [x)]]
(27)
(1 -
+ - J
Supposing that Ja is Lipschitz continuous and bounded, and fr is bounded, the following bound of enr is obtained by integration,
/ (I.(())(())) (z) (z)] + J [Io__u)) - =2|dss
leDI (f)] &
(28)
+ /" [[I (()) + S [|E(u) Ja(z)) (z)lld,
40
18

---

Published in Transactions on Machine Learning Research (02/2026)
where X denotes inequality up to constant factors. Therefore, combining Eqs. full error bound of the LLaSDI approximation
18
21 and
28
gives the
||({)]| = (() + edI(t))l
[vo(u) =/|ds ||eA [
+ / ||(I)(u)) J(u)
(29)
t0
[Jo(u)ú fz))] + - (z)2ss
To relate this to the training loss (Eq.
66] in tLaSDI, the following error terms can be defined
Eint to)
(1) =
to
Eree (1: || ||xxm())]
(30)
GJac (t: 1o) = Lo
Emod (t: [Jo(w)u (z) + +l Ja(z)ff (z)]lds.
It is clear that the method error now bounded as
||(()]| $ (t; + (ree to)
CJac (t: to) + Gmod to):
(31)
which are continuous counterparts to the loss terms defined previously. For example, applying the trapezoidal rule to approximate the integration within a small time interval (nt1) gives the following approximation to the dynamies integration error,
11+1
At (1) - =/|ds 2 110 2(tn+1)|l
Eint (I++1 tn)
fn
(32)
At 2
1.mm M (z((s)))ds
Φ (u((n++))) - (u.(u(t))
which corresponds to the integration loss Cint in Eq. Following a similar argument, all loss terms in Eq. O can be recovered from the corresponding error components in Eq. (30). This error estimate establishes theoretical upper bound the ILaSDI approximation error, thereby providing a theoretical foundation for the training strategy based on the loss function in Eq.
A.2 Effects of Jacobian and Modeling Loss Terms
The regularization parameters (weights) in the loss function in Eq. (6) are selected to ensure balanced contributions from each loss term. In the example of IDIV Plasov-Poisson Equation, it is observed that the Jacobian and modeling loss terms have relatively large magnitudes, as shown in the training loss history in Fig. a). To balance their influence on the total loss, Ajac = 10-9, and Amod = 10-6 are adopted.
To further investigate the effects of these loss terms on model performance, Asac is varied from 10-10 to 10-3, 3, while keeping the other weights fixed. An analogous study is performed for Xmad- As shown in Fig. b), the final values of LJcc and Lmod decrease monotonically with increasing AJac and Amod, respectively, This behavior is expected, as larger regularization weights impose stronger penaltics on the corresponding loss terms, resulting in smaller loss values. Fig. I(c) presents the average maximum relative error over the entire parameter space for models trained with different regularization weights. The results indicate that excessively strong or weak regularization associated with the Jacobian and modcling loss terms can adversely affect training and model generalization performance.
19

---

Published in Transactions on Machine Learning Research (02/2026)
10a
Erotal
0.055
Cint
0.07
Loss
Eriod
Coc
0.045
0.06
10-1
0.040
0.05
0.035
Apa
10
0.030
Amod
0.0
0.4
0.8
1.0
1.2
1.4
10-10
10
10-
-0
10
10
10
Epochs
1e5
(a)
(b)
Average Max. Relative Error (%)
3.00
Amod
Ajac
2.75
2.50
2.25
2.00
1.75
10-10
10-9
10-8
10-7
10-6
10-3
10-4
-0
Figure 9: 1D1V Vlasov-Poisson Equation (a) Training loss history; (b) Relationships between L and AJac and between Emod and Amodi (c) Effects of the regularization parameters (AJacs Amod) on prediction accuracy.
20

---

Published in Transactions on Machine Learning Research (02/2026)
A.3 Speed-up Analysis
For the IDIV Vlasov-Poisson Equation, the proposed LLSDDI model achieves an average error of 1.66%, reported in Section 3.2 The training data were generated by using high-fidelity simulations run on 4 cores of an Intel Xeon Platinum 8480 CPU. Training the tLaSDI model on an NVIDIA RTX 4000 Ada GPU takes around 45 minutes. For comparison, a single numerical simulation that has a similar error level (1.66%) with respect to the training data, produced by decreasing the spatial resolution of the high-fidelity model, requires 9.58 seconds, whereas the inference time of the trained tLaSDI model is 0.00384 seconds, yielding a 2,495x speed-up.
The time step size used to generate the training data is 5 x 10-3, Increasing the time step size of the high- fidelity simulations to 10-2 and 2 x 10-2 while keeping the spatial resolution fixed reduces the computational time to 5.8 seconds and 5.26 seconds, respectively, and incurs small relative errors of 0.0004% and 0.014% with respect to the training data computed at the smaller time step size. Further increasing the time step size leads to unstable simulations and convergence issucs. When comparod to the high-fidelity simulation with the largest time step size of 10-2, the proposed (LaSDI model achieves 1,370x speed-up. Although the examples in this paper rely on simulated data generated from PDE solvers, we emphasize that the proposed tLaSD1 model is equally applicable to settings involving dynamic sensor data where the governing ODEs/PDEs are not known a priori.
A.4 PGFINN Experiments
In this section, demonstrate the performance of the proposed pGFINN by applying it to common bench- marks: two gas containers and two-mass thermo-mechaniee system.
A.4.1 Two Gas Containers
The first experiment of pGFINN involves two ideal gas containers separated by a movable wall that allows heat and volume exchange. The evolution of the state variables (g.P, S1, is governed by the following ODE system,
p j
m
1 =
E
Eg
2-4
(33)
$1 = π
a
T. T2
& $2 = Ta
T5 Ti
where is the wall mass, 9 resp. p are the position resp. momentum of the movable wall, T, F and S2 resp. E,, i == 1.2, are the entropy resp. energy of the two subsystems, determined from the Sackur- Tetrode equation (Schroeder 2020) for ideal gases, S:/NB = In (2V,E) Here, V1 = q and V/2 2 q are the volumes of the containers, N is the number of gas particles, and kg is the Boltzmann constant. In this 1 is adopted. The total entropy of the system is S S1 + S2, and the total energy
experiment, Nkg of the system is F- = /2m(m)) F, + Eg. This prior knowledge can be embedded into PGFINN by defining ENN := E and SNN == E, as described in Section 2.1
We first consider a system parametrized by a E [1,50]. The training data is generated using the fixed initial condition (0.87, 0.44, 1.00, 1.60) with m 1, while varying a within the specified parameter range. Fig. 10 a) compares the maximum relative crrors of pGFINN trained with varying numbers of uniformly distributed sampling points in the parameter space. Increasing the number of training points from 2 to 7 reduces the error from 17.0% to 3.3% demonstrating the effectiveness of pGFINN parameterization. However, the model continues to exhibit larger errors in regions with smaller values, and further reducing these errors using uniform sampling would require substantially more training points.
On the other hand, by integrating PGFINN with the physics-informet informed active learning strategy described in section [.. the PGFINN model intelligently selects training points in high-crror regions. As shown in Fig. 10(b), where red circles resp. blue stars denote training points selected via uniform resp. adaptive sampling, the maximum relative error is significantly reduced to 1. 1.2% with adaptive sampling, compared to 3.3% with
21

---

Published in Transactions on Machine Learning Research (02/2026)
uniform sampling. Moreover, the distribution of errors over the range of a is substantially homogenized with active learning, ensuring that no values of a are unexpectedly erroneous relative to the others. Notably, the active learning strategy achieves this improvement using only 6 training points, fewer than the required by uniform sampling, highlighting its efficiency and effectivenoss.
As another experiment, we increase the complexity of the two gas container system's parameterization by fixing a = 10 and considering m [0.5,5] which has nonlinear relationship with the state variables. Fig. (0'c) shows that PGFINN with adaptive sampling achieves significant reduction in the maximum relative error, from 12.8% (with 9 uniformly distributed training points) to 1.9%, using only 8 adaptively selected sampling points. Again, it can be scen that the errors are dramatically uniformized with adaptive sampling, illustrating its strong performance even in this more difficult case.
17.5
Max Reietive (80
Train Points
3.0
Adaptive
Max Relaove Eroo 15)
10-
Accptivv
12.5
Points
2.5
00.0
2.0
0
7.5
Reaatvv
1.5
6
5.0
1.0
4
2.5
0.5
2
0.0
30
35
40
45
15
25 30
45
50
0.5
1.0
1.5
2.5
2.5
3.0
3.5
4.0
4.5
5.0
Alpha
(a)
(b)
(c)
Figure 10: Two gas containers (a) Maximum relative errors for a parameterization with varying numbers of uniformly distributed training points; (b) and (c) comparison of the maximum relative errors between uniform and adaptive sampling for a and m parameterization, respectively. Red circles and blue stars denote training points selected via uniform and adaptive sampling, respectively.
We further evaluate PGFINN under 2D parameterization by considering both [5, 20] and E [0.5,,) 3]. As shown in Fig. I1 PGFINN with just 14 adaptively selected training points significantly reduces the maximun relative error to 3.2% 3.2%, compared to 21.1% with 25 uniformly distributed training points. Notice that the red vertical band in (a) of parameter combinations with high maximum relative error is effectively mitigated in (b) despite using many fewer training points, further highlighting the offectiveness and advantage of the active learning strategy described Section 2.3
A.4.2 Two-Mass Thermo-Mechanica System
To continue testing PGFINN, we further increase the complexity of our experiments by considering a two- mass thermo-mechanical system with damping and heat exchange. The evolution of the state variables (g) 42, Pl, P2, S2, is governed by the following ODE system,
d1 P1
my
d2 m2
P2
P1 k(q1 (2) a
m1
mg
(34)
P2 = - k(g2
91) + a
Pi
P1
Ši 2T1
a PL
P1
+ 3
T5
T
S2 212
PL
82
3
)
where mi, i = 1,2, denotes the i-th mass, resp. pi are the position resp. momentum of the i-th mass, and E, resp. S, are the energy resp. entropy of the i-th mass, with E, = c.S; and T: = 0E;/0S; C. Further, & is the spring constant between masses, a is the damping coefficient, and 3 is the heat conductivity coofficient. The total entropy of the system S = S₁ + S2, and the total energy of the system is E = p2m(2m1) + (2m2)) k/2(q1 + 92)2 Ei + Ez. The following experiments adopt the choices m, = G = 1,2.
22

---

Published in Transactions on Machine Learning Research (02/2020)
5.00
08
21
R8
05
6.7
00
0.7
0.
9-
044
0.
00
06
0.
5.00
09
k.
1.
1
06
05
0
O
27
05
04
04
06
0.75
04
20
84
300
0.5
0.6
0.7
34
0.3
0.4
077
05
0.0
09
1.0
1.4
0.78
1.8
1.3
1.0
0.7
0.
0.5
0.
0.7
26
0.4
0.
06
08
0.6
0.0
0.7
08
06
0.
600
20.0
6.00
1.4
25.0
735
725
8000
05
17
90
0.5
02
C3
03.08
8.00
16
14
08 оa os
07
GO 1 07 04
02
09
20
21
20
12 08
17.5
8.7
R3
16
9
R.6
16
DUG
02
02
02
03
68
96
05
8.75
1.3
1.3
0
oG
05
B7
CO
14
24
02
00
13
17
20
21
23
15
6.50
03
15
92
0.2
1.7
0.7
0.
02
02
04
06
9.50
1.0
1.3
06
04
0.5
08
06
34
0.2
07
12
1.8
20
18
10:25
0.3
0.2
02
0.
10:25
27
05
24
03
1.
Alpha
11.75
0.2
02
02
02
12.5
Alpha
11.75
57
04
0.4
0.5
1.6
08
12,00
0.2
02
2.
26
OU7
02
0.3
0.3 02
04
1.0
1.2 1.3
9.7
12,50
3/8
1.7
0.9
04
06
0.7 0.0
02
23
066 099 1.2
1.5
1.5
06
13:25
82
14
022
25
2
02
03
03 02
UA
CF
1.1
12
1.9
0.F
-10.0
19:25
1.0
1.5
09
04
OF
0.
CB
US 04 24 B7
09
12
14
1.5
1.5
15
11
16.00
02
039
02
03
03
02
03
ON
0M
14.00
1.
2:
0a
08
11 TO
04
24
07 O8
12
1.
15
11
-75
15.50
n2
04
02
0
02
14
07
08
1
07
16.25
02
68
04
3.8 34
2:2
1.1
03
04
02
02
66
98
06
04
-50
16.25
1.1
22
1.0
0G
12
14
12
0
04
DE
06
00
1
10
1.1
111
11
05
06
17:00
01
8:0
05
3.5
36
2.3
1.2
0.5
05
04
02
02
65
25
08
07
03
17.00
1.0
2.
10
05
13
1.
11
4
24
28
10
11
1.1
10
0.0
09
09
08
07
05
17.75
01
08
03
17.75
58
13
15
03
02
10.30
02
10.30
20
19:25
09
0.5
08
19.25
23
1.5
20.00
07
88
14
05
02
02
88
96
02
20.00
05
14
08
18
08
29
25
08
5 0000000
060
19
m
m
(a) Uniform
(b) Adaptive
Figure 11: Two gas containers Comparison of the maximum relative errors across the parameter space between (a) uniform sampling with 25 training points and (b) adaptive sampling with 14 training points. The training points are highlighted with black boxes.
We first consider a system parametrized by a E [0.1, 1]. The training data is generated using the fixed initial condition (4.98, 0.04, 0, 9.96, 1.93, 1.92) with h = 10, while varying a within the specified parameter range. As shown in Fig. a)) increasing the number of training points from to 8 reduces the error from 53.9% to 8.5% again illustrating the impact of the PGFINN parametorization, However, uniform sampling suffers from the same drawback as before, and the model continues to exhibit larger errors in regions with smaller a values, which correspond to stronger oscillatory dynamics. Further reducing these errors using uniform sampling would require substantially more training points.
In contrast, Fig, b) shows that PGFINN with active learning intelligently selects training points in high- error regions. With only 7 adaptively selected training points, the proposed adaptive sampling procedure reduces the maximum relative error to 4.1%, compared to 8.5% achieved by uniform sampling with 8 points.
Fixing a 0.1, we then consider a different parameter k E [0.5,30] Fig. 12(c) shows that pGFINN with adaptive sampling achieves significant reduction in the maximum relative error in this case as well, from 2.8% (with 8 uniformly distributed training points) to 1.0% using only 7 adaptively selected points. Moreover, it can be seen that adaptive sampling has again substantially uniformized the maximum errors over the parameter range, guaranteeing consistent dynamical predictions regardless of the value of k.
We further evaluate pGFINN under 2D parameterization by considering both a E [0.1, 1] and & E [0.5, 10). As shown in Fig. 13 PGFINN with 28 adaptively selected training points significantly reduces the maximum relative error to 4.4%, compared to 23.6% using uniform sampling with 30 training points. Similarly to the two gas containers case, the horizontal band in (a) of high maximum crrors has been climinated in (b) due to the proposed active learning stralegy. This result further demonstrates the elfectiveness of this procedure in improving the generalization performance of pGFINN dynamical predictions to unseen parameter configurations.
A.5
Implementation Details
This scction summarizes the implementation details of the experiments presented in Scction 3.1 3.2 and
3.3 The high-fidelity simulations were performed on an IBM Power9 CPU with 128 cores and 3.5 GHz. All training and testing in the experiments were conducted using NVIDIA A100 GPU with 40GB of memory.
23

---

Published in Transactions on Machine Learning Research (02/2026)
30
Irain No eTS Train Po ms
Unitorm Adaptive
Uniform
2.5
Adaptive
60-
Reletive Croo
2.0
Max Relative
1.5
20
Max
100
0.0
0.2
0.4
0.7
1.0
0.2
Alpha
(a)
(b)
(c)
Figure 12: Two-mass thermo-mechanical system (a) Maximum relative errors for a paramctcrization with varying numbers of uniformly distributed training points; (b) and (c) comparison of the maximum relative errors between uniform and adaptive sampling for a and k parameterization, respectively. Red circles and blue stars denote training points selected via uniform and adaptive sampling, respectively.
0.10
8.6
1:0
1.0
0.5
04
04
0.5
0.5
86
36
06
0.4
0.4
044
0.10
8.7
8.6
1.5
1.2
0:15
014
11
12
13
14
17
18
19
20
21
21
8115
1.5
18
0:1
13
M
20
21
20
0.19
84
0.4
264
1.3
8.0
0.7
88
0.42
04
0.5
0.7
0.
088
0.5
02. 32
0.7
0.42
1.5
1.1 c88
02 02 06
0.45
0.2
3.5
0.6
088 0.9 63
0.9
0.5
B4
24
C/4
04
0.6
0.6
0.8
8.6
18
0.46
06
0.6
c.6
02
0.2 03 0.4 044
06 0.5 3.
15
Alpha
0:51
02
944
us
0v U. 04
955 UB
Ub 05
04
34
04 U. 03
Alpha
0:51
0.5
0.1
us
U. US 04 04 04 04 LA
U3 01 D1
0.55
0.3
3.5
0.6
06
0.5 06
0.G
0.5
0.6
36
04
0.4
0.50
1.4
0.
c8
0a
07
0.6 0G 06 07 0G 06
0.4
0.0
0.0
0.6
32
0.2
0.2
0.77
06
0.7
966 05
05
0.
0.77
42
40
0.4
02
05 04 04
012
03
R6
07
96 86 84
D. 06
0. B6 06
86
0.
4.0
35
36
13
07
07 08
OE
0F
07 06
03
0.88
044
044
06
06
05
0.5
0.5
0.5
0.5
0.4
0.88
85
35
32
10
1.0
1.1
1.5
081
03
08
88
05
044
0:1
0.4
081
23
23
22
1.8
36
10.00
k
(a) Uniform
(b) Adaptive
Figure 13: Two-mass thermo-mechanin system Comparison of the maximum relative errors across the parameter space between (a) uniform sampling with 30 training points and (b) adaptive sampling with 28 training points. The training points are highlighted with black boxes.
24

---

Published in Transactions on Machine Learning Research (02/2026)
The Adam optimizer Kingma & Ba 2014) from PyTorch Paszzk et al. 2019 was used for model training. All codes and data to regenerate the results in this paper can be found in the GitHub repository (https:
//github.comm
http://///oooogg///ooooaaaaaa -NNN-LLLDT) This research makes use of the following open-source assets:
http://aw...ccopppssss/////aaa/a LLSDDI (Park et al. 2024) (MIT license, http://///ooooooeeeeeeeeerrrrrrr
fourier neural operator
(MIT license,
http://hhabcc/aaaaaaoooooa/rrera http:///ooorrreeurraaaaaaea cacrrrr_eurrrll operator). deepppoeee,, deeponet-Ino (Lu el al. 2021
Li et al. 2020 Lu et al. 2022) BCCNY-N 4.0 license, httpss:s/,ttts:///// htt.:///h......ooooo..a////aaata
and HyPar (MIT license, httpa://github... http:/thhw...mmmmbbb//paaaaaaa http:://gghhaaaaocooororrggrg/gp
A.5.1 One-dimensional Burgers' Equation
This section summarizes the implementation details corresponding to the results in Section 3.1
Data Generation High-fidelity simulations employ a uniform spatial discretization with nodal spacing Ar = 6/1000 and the backward Euler method with At = 10 for time integration. The solution data are gencrated using an in-house numerical solver and the derivative of the data is computed using a backward difference scheme. The training data is subsampled from this dataset to have spatial dimension of 200 and temporal dimension of 201.
The encoder adopts a 200-100-5 architecture with ReLU activation functions, while the decoder employs a symmetric structure. The hypernetwork contains 3 layers, with 20 neurons in each hidden layer and hyperbolic tangent activation. All NNs in GFINN and pGFINN contain layers, with 40 neurons in each hidden layer and hyperbolic langent activation. The forward Euler method is employed for the time integration latent dynamics.
Model Architectures
Training The training conducted using the loss function in Eq. G with regularization parameters set as Arec 10-1, Asac 10-9, and Amod = 10-7 The initial learning rate is set to 10-4 and decayed by a factor of 0.99 every 2,000 epochs. The models are trained for a total of 15,000 epochs with batch size of 50. The tLaSDI model, integrated with the physics-informed active learning strategy, uses update epoch Nup =3,000 i.c., adaptive sampling is performed every Nup epochs. Training begins with parameter points located at the corners of the parameter space, and 4 additional parameter points are adaptively selected during training.
A.5.2 1D1V Vlasov-Poisson Equation
This section summarizes the implementation details corresponding to the results in Section 3.2 and 3.3
Data Generation High-fidelity simulations employ WENO spatial discretization (64 64) Jiang & Shu 1996) and a fourth-order Runge-Kutta explicit time integration scheme with A/ = 5 x 10-3 The solution data are generated using the HyPar solver (Ghosh 2025) and the derivative of the data is computed using a backward difference scheme. The training data is subsampled from this dataset to have a spatial dimension of 1,024 and a temporal dimension of 251.
LLSSDI Architecture and Training The encoder adopts a 1.024-200-100 architecture with ReLU activation functions, while the decoder employs a symmetric structure. The hypernetwork contains layers, with 20 neurons in each hidden layer and hyperbolic langent activation. All NNs in GFINN and pGFINN contain 5 layers, with 40 neurons in each hidden layer and hyperbolic tangent activation. The forward Euler method is employed for the time integration of latent dynamics.
The training is conducted using the loss function in Eq. 6 with regularization parameters set as Avec 10-1, AJae = 10-9, and Amod = 10-6 The initial learning rate is set to 10-4 and decayed by a factor of 0.99 every 2,000 epochs. The tLaSDI models are trained for 50,000 epochs with a batch size of 50, followed by an additional 100,000 epochs with a batch size of 400.
DeepONet Architecture and Training In the benchmark comparison (Section 3.3 DeepONet consists of branch network and trunk network, each with 1,024-256-256-166 168 architecture and ReLU activations.
25

---

Published in Transactions on Machine Learning Research (02/2026)
The model is trained using mean squared error loss over 200,000 epochs, with an initial learning rate of 10-3 and weights initialized via the Glorot (Xavier) normal scheme. The learning rate is scheduled to decay according to an inverse time decay rule: 10 10-3/(1 + 10 x iteration).
FNO-3D Architecture and Training In the benchmark comparison (Section 3.3 FNO-3D is configured with 11 channels, retains the first 5 Fourier modes, and uses ReLU activations. The model is trained using a relative lz loss over 1,000 epochs, with an imitial learning rate of 2.5x10-3 that is halved every 100 epochs. The input data is normalized by using unit Gaussian scaling and weight decay regularization of 10-4 is applied to the loss function.
26


Reviews:
[{"review_id": "DKRtvr8ofk", "reviewer": "Reviewer_RFRH", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The Manuscript describes a physics informed structure the describe Hamiltonian (Poisson brackets) for energy conservations and symmetric propagator  for entropy production. The architecture uses an auto-encoder to find a latent representation  that has  a lower rank (dimension) to describe a higher complexity system.  This way a simpler, more robust  model can be used and at the same time the basic metriplectic (the one that keeps the thermodynamic 1s and second laws t (energy conservation 1st law) and 2nd law (positive entropy production) valid.\n\nThe architecture is validated with Burgers equation that has non-linearity forcing a shock wave that is regularized with friction and a more complex Vlasov-Poisson model for plasma physics, where the particles have repulsive interactions.  The results are promising, the code runs faster and its retains its accuracy.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The experiments with the architecture provide a convincing  and clear evidence by producing a lower dimensional latent representation of the essential physics of thermodynamical systems out of equilibrium driven to maximal entropy and the final equilibrium state.\n\nThe text is clear, the graphics supports the narrative, and the example cases are well chosen to illustrate the strength of the physics informed architecture.", "audience_yn": "Yes", "audience_explanation": "The Authors are providing an architecture that uses lower rank representation of a dynamic systems where basic physical laws are enforced. One can see how the neural network learns \"by heart\"  the principal features of the solution and its dynamics for the particular Hamiltonian and friction components rather than solving the equation in a generic manner - and as expected - this particular, \"specialized\" model can be described by a much smaller latent space than the degrees of freedom of the problem would indicate.", "requested_changes": "I like the Manuscript as it is.", "broader_impact_concerns": "I do not have concerns on the ethical implications.", "additional_comments": "I support acceptance of this Manuscript as being informative and interesting,  but also as a step to more complicated, higher dimensional models that would naturally require much more computing power. Machine learning is not always about who is having the biggest machines, but also about who has the good, creative ideas.", "extra_scores": {}, "extra_text": {}}, {"review_id": "ntP64QMTSo", "reviewer": "Reviewer_puUj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes an enhanced \"thermodynamics-informed latent space dynamics identification\" (tLaSDI) framework for building reduced-order models (ROMs) of parametric nonlinear dynamical systems. The core idea is to learn a low-dimensional latent representation of a high-fidelity system whose dynamics are not only accurate but also inherently obey the first and second laws of thermodynamics (energy conservation and non-decreasing entropy).\n\nStrength:\n- The motivation is clear and the paper is well-organized.\n- The most significant strength is the rigorous embedding of thermodynamic priors (via the GENERIC formalism) directly into the neural network architecture. This is a substantial advantage over \"black-box\" operator learning methods (like DeepONet or FNO), as it guarantees physically plausible long-term behavior (e.g., stability) and provides interpretability.\n\n\nWeakness:\n- A potential limitation for real-world applicability is the reliance on the derivative data (u_dot) for constructing key terms in the loss function (L_Jac, L_mod). The paper briefly mentions that a weak-form formulation could be used for noisy data, but this is not explored in the experiments. \n- The test cases, while standard, are relatively low-dimensional in their parametric and spatial complexity (2-3 parameters, 1D/1D1V problems). It remains unclear how well the method would scale to problems with higher-dimensional parameter spaces (e.g., >5 parameters) or much higher-dimensional physical states (e.g., 3D CFD simulations).\n- Although Appendix A.2 briefly explores the sensitivity to the weights (λ_Jac, λ_mod) of the more complex loss terms, a more comprehensive ablation study would strengthen the paper. For instance, quantifying the individual contribution of the Jacobian loss (L_Jac) and the modeling loss (L_mod) to the final model performance would provide deeper insight into the necessity of each component.", "claims_and_evidence_yn": "No", "claims_explanation": "See Weakness. The major concern is how well the method would scale to problems with higher-dimensional parameter spaces (e.g., >5 parameters) or much higher-dimensional physical states (e.g., 3D CFD simulations).", "audience_yn": "Yes", "audience_explanation": "The proposed tLaSDI framework replaces a complex \"hyper-autoencoder\" from prior work with a standard autoencoder coupled with the new pGFINN. This simplifies the parameterization, leading to reductions in model complexity, training cost, and inference time. Therefore, the findings might be helpful for other research.", "requested_changes": "Conduct additional experiments to clarify the weaknesses.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "sXhMHaxHat", "reviewer": "Reviewer_KFzk", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this work the authors explore improvements to a physics aware scheme for dimensionality reduction of ODE (and therefore discretized PDE) systems. The method combines an autoencoder setup with a neural network approximation scheme that respects the first two laws of thermodynamics (pGFINN). Part of their contribution is showing that a simple autoencoder setup is sufficient.\n\nThe authors define a multi-component loss for this setup. They also define an active learning scheme to improve training performance. They then benchmark this approach on 1-D Berger's Equation and Vlasov-Poisson, for ~ 200 steps. They show that the relative error is around 1%, and show that this is better than previous methods. The model also trains faster than the approach it is directly derived from.", "claims_and_evidence_yn": "Yes", "claims_explanation": "To the best of my ability to check and understand them, the experiments are sound. The text in the paper is supported by the figures and tables. I am not an expert in this area so I will defer to other reviewers on whether or not the chosen benchmarks are the correct ones.", "audience_yn": "Yes", "audience_explanation": "The area of PDE approximation via dimensionality reduction + NN based function approximation is active, and this paper will be interesting to people in that field. The paper is also well written and can serve as an introduction to the basic approaches, particularly methods to put physical constraints/symmetries/conservation laws into the model.", "requested_changes": "Does the 3000x speedup include the training time of the model? Alternatively, how does the training time of the model compare to single simulations? This information is important to understand the scenarios where one may want to deploy this method --- e.g. if training is expensive, then this method would be less useful for projects where many different models have to be tested and compared with each other.\n\nRelated to the question of speedup: what is the speedup compared to a standard ODE/PDE iteration scheme with similar error properties? Numerical integration can be sped up by choosing larger discretization in space or time, at the cost of more errors. What is the runtime of the proposed method compared to a traditional method with more coarse timesteps? To me this is an important comparison to properly position the method against classical approaches. **I think this comparison is vital for the paper.**\n\nHow does the proposed method work on more chaotic systems? For example KdV? In this setting the relative error may be larger, but it would be interesting to know if the macroscopic physical phenomena are still preserved (e.g. soliton solutions). An example like this might make the paper stronger.", "broader_impact_concerns": "No.", "additional_comments": "I am not an expert in the area of PDE approximation; the benchmarks look good to me but I will defer to more knowledgeable reviewers on this point.", "extra_scores": {}, "extra_text": {}}, {"review_id": "hSpgExkD5V", "reviewer": "Reviewer_91jd", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a regularization scheme for the latent space of an autoencoder that encourages compliance with the domain knowledge of thermodynamics. This scheme consists in:\n-an active learning method for focusing relevant training points instead of randomly sampling uniformly\n-two regularization functions based on physically relevant constraints\n-latent dynamics that are compliant with domain knowledge (physics) and provide interpretability?", "claims_and_evidence_yn": "Yes", "claims_explanation": "First of all, I want to mention that I am not an expert of PINNs nor in physics, making my competencies in evaluating this paper quite limited.\n\nNonetheless:\n-the model's performance is demonstrated by the good reconstruction quality shown in table 1.\n-the evaluation is accompanied by a relevant ablation study showing the positive effect of the adaptive sampling.\n\nI think that these should be extended to show the effect of the regularization terms on the latent representation's compliance with the physical constaints. How do the weights of the jacobian and modeling error affect the outcome? Does the reconstruction become better on average, but sees more anomalies? This would be crucial to measure the impact of that physical information.", "audience_yn": "Yes", "audience_explanation": "This paper should be relevant for the PINNs community.", "requested_changes": "Deep study of the isolated impact of each element (active learning is treated, but what about the regularization scheme?).", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)