Domain-Decoupled Physics-informed Neural Networks with

Closed-Form Gradients for Fast Model Learning of Dynamical Systems

Henrik Krauss

1,2 a

, Tim-Lukas Habich

2 b

, Max Bartholdt

2 c

Thomas Seel

2 d

and Moritz Schappler

2 e

Department of Advanced Interdisciplinary Studies, The University of Tokyo, Tokyo, Japan

Institute of Mechatronic Systems, Leibniz University Hannover, 30823 Garbsen, Germany

{tim-lukas.habich, max.bartholdt, thomas.seel, moritz.schappler}@imes.uni-hannover.de

Keywords:

Physics-Informed Machine Learning, Surrogate Modeling, Model Learning, System Dynamics.

Abstract:

Physics-informed neural networks (PINNs) are trained using physical equations and can also incorporate un-

modeled effects by learning from data. PINNs for control (PINCs) of dynamical systems are gaining interest

due to their prediction speed compared to classical numerical integration methods for nonlinear state-space

models, making them suitable for real-time control applications. We introduce the domain-decoupled physics-

informed neural network (DD-PINN) to address current limitations of PINC in handling large and complex

nonlinear dynamical systems. The time domain is decoupled from the feed-forward neural network to con-

struct an Ansatz function, allowing for calculation of gradients in closed form. This approach signiﬁcantly re-

duces training times, especially for large dynamical systems, compared to PINC, which relies on graph-based

automatic differentiation. Additionally, the DD-PINN inherently fulﬁlls the initial condition and supports

higher-order excitation inputs, simplifying the training process and enabling improved prediction accuracy.

Validation on three systems – a nonlinear mass-spring-damper, a ﬁve-mass-chain, and a two-link robot –

demonstrates that the DD-PINN achieves signiﬁcantly shorter training times. In cases where the PINC’s pre-

diction diverges, the DD-PINN’s prediction remains stable and accurate due to higher physics loss reduction or

use of a higher-order excitation input. The DD-PINN allows for fast and accurate learning of large dynamical

systems previously out of reach for the PINC.

1 INTRODUCTION

Physics-informed neural networks (PINNs) have been

introduced by (Raissi et al., 2019) as a machine-

learning framework to solve ordinary (ODEs) and

partial differential equations (PDEs). On top of su-

pervised learning of a feed-forward neural network

(FNN) on data, PINNs introduce a custom physics

loss based on the system-governing ODEs or PDEs

including boundary/initial conditions. This seamless

integration of physical laws into the neural-network

training process leads to accurate predictions even

with limited data as well as the ability to extrapolate

on out-of-distribution (unseen) data, while serving

as a fast surrogate model. So far, many variants

https://orcid.org/0000-0002-9787-5465

https://orcid.org/0000-0003-4167-8443

https://orcid.org/0000-0002-8422-9368

https://orcid.org/0000-0002-6920-1690

https://orcid.org/0000-0001-7952-7363

of PINNs have been successfully applied to ﬁelds

such as thermodynamics, chemistry, material science

(Karniadakis et al., 2021; Cuomo et al., 2022), and

dynamical systems, including robotics (Hao et al.,

2022). PINNs have been adapted for the control of

dynamical systems (PINCs) by (Antonelo et al., 2024)

where the FNN predicts the system’s evolution over a

short time interval assuming constant excitation. This

concept is compatible with and has been applied to

state estimation using an Unscented Kalman Filter

(UKF) (de Curt

o and de Zarz

a, 2024), model pre-

dictive control (MPC) of robotic systems (Nghiem

et al., 2023), such as two-link manipulators (Nicode-

mus et al., 2022; Yang et al., 2023) and quadro-

tors (Sanyal and Roy, 2023). Here, the PINC out-

performs traditional numerical integrators regarding

prediction speed of the system dynamics.

However, the training of PINNs can be challeng-

ing and many efforts to improve trainability have been

developed (Wang et al., 2023). These include adap-

Krauss, H., Habich, T., Bartholdt, M., Seel, T. and Schappler, M.

Domain-Decoupled Physics-informed Neural Networks with Closed-Form Gradients for Fast Model Learning of Dynamical Systems.

DOI: 10.5220/0012935200003822

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 21st International Conference on Informatics in Control, Automation and Robotics (ICINCO 2024) - Volume 1, pages 55-66

ISBN: 978-989-758-717-7; ISSN: 2184-2809

tive activation functions (Jagtap et al., 2020), various

loss-balancing techniques such as GradNorm (Chen

et al., 2018), SoftAdapt (Heydari et al., 2019), learn-

ing rate-annealing (LRA) (Wang et al., 2021), rel-

ative loss balancing with random lookback (ReLo-

BRaLo) (Bischof and Kraus, 2021), dynamically

normalized PINNs (DN-PINNs) (Deguchi and Asai,

2023), adaptive collocation point sampling strate-

gies (Wu et al., 2023), as well as training on non-

dimensional PDEs (Kapoor et al., 2024). Also, for

dynamical systems with chaotic behaviors, loss func-

tions might need to be reformulated to respect spatio-

temporal causality to ensure training convergence

(Wang et al., 2022).

One main point of research for PINNs is the topic

of differentiation of the FNN output with respect to

the spatial and temporal domain inputs, i.e., the inde-

pendent variables in the ODE or PDE over which the

solution is deﬁned. Differentiation is required to eval-

uate the ODE/PDE and determine the physics loss but

is very computationally expensive. Training times of

the PINC (Antonelo et al., 2024) can amount to sev-

eral days or weeks, rendering the learning of large or

complex dynamical systems unfeasible. Here, previ-

ously developed architectures can be summarized in

the ﬁve categories of using

(i) automatic differentiation (AD),

(ii) numeric differentiation,

(iii) a hybrid of (i)–(ii),

(iv) a variational approach, or

(v) closed-form gradients.

(i) Classical PINNs as formulated by (Raissi et al.,

2019) or (Berg and Nystr

om, 2018) use graph-based

automatic differentiation (a-PINNs) as implemented

in PyTorch or TensorFlow to determine the FNN

gradients. However, AD is computationally expen-

sive and a-PINNs can only reach high accuracy for

large numbers of collocation points. (ii) PINNs

based on numerical differentiation (n-PINNs) avoid

AD by using numerical differentiation on the collo-

cation points. This makes them more robust to a low

number of collocation points but may attain lower ac-

curacy than a-PINNs (Cuomo et al., 2022). (iii) Ad-

dressing the disadvantages of both approaches, (Chiu

et al., 2022) have used a coupled-automatic-numerical

(CAN) differentiation method and introduced CAN-

PINNs – a hybrid approach between a- and n-PINNs.

Their residual loss is formulated at locally adjacent

collocation points through a Taylor-series scheme that

includes derivatives at the collocation points obtained

from automatic differentiation. However, they require

the selection of an appropriate numerical scheme and

therefore cannot be straightforwardly interchanged

with a-PINNs. (iv) Another approach are variational

PINNs (v-PINNs) introduced in (Kharazmi et al.,

2019). Here, a variational loss function is constructed

within the Petrov-Galerkin framework. For that, the

PINNs’ output is numerically integrated and tested in

a test space of Legendre polynomials. Variants of the

v-PINN have been developed that divide the domain

into subdomains (Kharazmi et al., 2021; Liu and Wu,

2023) (v) Also, it is possible to compute analytical

gradients without graph-based AD, such as for sin-

gle hidden-layer networks (Lagaris et al., 1998). Al-

ternatively, the FNN does not predict the solution to

the ODE or PDE directly but parameters of basis or

Ansatz functions that approximate the solution. Ex-

amples using this approach are:

• Taylor PINN (Zhang et al., 2023),

• Polynomial-interpolation PINN (Tang et al.,

2023),

• Legendre-improved extreme learning machines

(Yang et al., 2020), and

• Multiwavelet-based neural operators (Gupta et al.,

2021).

In the context of PINNs for control, to the best of

our knowledge, only AD-based architectures such as

the PINC have been applied (Antonelo et al., 2024),

featuring the assumption of constant excitation in the

short prediction horizon. Three main limitations of

this PINC for the learning of large and nonlinear dy-

namical systems are:

(I) Graph-based automatic differentiation causes

long training times. The number of AD opera-

tions 𝑛

per epoch is proportional to the num-

ber of system states 𝑚 and the number of col-

location points 𝑛

collo

, where 𝑛

collo

itself may be

chosen higher for larger or more complex sys-

tems.

(II) Constant excitation (zero-order-hold assump-

tion) for longer trained time intervals results in

poor accuracy. However, longer time intervals

are advantageous for fast simulation since one

prediction with the surrogate model should sim-

ulate a large time step with high accuracy.

(III) Initial-condition (IC) loss and physics loss can

require a weighting scheme such as Grad-

Norm, SoftAdapt, LRA or ReLoBRaLo. These

weighting schemes need to be chosen empiri-

cally, are sensitive to their hyper-parameter set-

tings, and can further increase training time.

For the prediction horizons used in control scenar-

ios, approximating the solution through differentiable

Ansatz functions to calculate gradients in closed-form

has the potential to drastically reduce training times

while maintaining high accuracy. For the ﬁrst time,

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

data

phys

data

Closed-form

gradients (fast training)

AD (slow training)

∗

𝑡

ˆx

𝑡

Δ ˆx

𝑡

SSM

𝑢

MSE

FNN

ˆx

𝑘+1

ˆx

𝑘+1

𝑘

𝑇

𝑘

𝑧

−1

𝑧

−1

∗

𝑘

𝑇

FNN

phys

data

𝜕𝑥

𝜕𝑡

ˆx

𝑡

SSM

FNN

MSE

𝜆

(a)

Training

DD-PINN

PINC

Self-loop prediction

(b)

Figure 1: Conﬁgurations for (a) PINC training and (b) PINC self-loop prediction process as well as (c) DD-PINN training

and (d) DD-PINN self-loop prediction. Time domain is decoupled from the FNN for the DD-PINN, enabling calculation of

gradients in closed form.

we apply the approach (v) of using analytical, closed-

form gradients for state-space learning of dynami-

cal systems. We propose a domain-decoupled PINN

(DD-PINN) architecture where the time domain is de-

coupled from the FNN to construct an Ansatz func-

tion that enables calculation of gradients in closed

form. It is formulated without being limited to a

speciﬁc Ansatz function, unlike previous work, and

can be employed interchangeably with the PINC for

physics-loss calculation during training as a state esti-

mator or in control such as for model predictive con-

trol. The DD-PINN is evaluated on three exemplary

dynamical systems with varying system complexity

and compared to the PINC (Nicodemus et al., 2022;

Antonelo et al., 2024) with regards to its limitations.

In Sec. 2, the fundamentals of the PINC are ex-

plained and the DD-PINN is formulated in Sec. 3.

The validation of the DD-PINN and a comparison to

the PINC follows in Sec. 4, preceeding the conclusion

in chapter 5.

2 PRELIMINARIES

In the original PINN for control (PINC) (Antonelo

et al., 2024), dynamical systems in form of a state-

space model (SSM)

x(𝑡) = f

SSM

(x(𝑡),u(𝑡)) (1)

are approximated using a neural network, where x de-

notes the state vector, 𝑡 the time domain and u an in-

put signal. The PINC predicts the future state

ˆx

𝑡

= f

,𝑡,θ) ≈ x(𝑡) (2)

with the feed-forward neural network f

continu-

ously over the sampling interval

0 ≤ 𝑡 ≤ 𝑇

(3)

in which the excitation is assumed constant and sub-

ject to boundaries

min

≤ x

max

min

≤ u

max

(4)

The parameters, i.e., the weights and biases θ of

the FNN are trained according to the process visual-

ized in Fig. 1(a) using a custom physics-informed loss

function

L(θ) = 𝜆

(θ)+𝜆

phys

(θ)+𝜆

data

(θ) (5)

Domain-Decoupled Physics-informed Neural Networks with Closed-Form Gradients for Fast Model Learning of Dynamical Systems

consisting of the initial-condition loss

= MSE(x

, ˆx

), (6)

physics loss

phys

= MSE



𝜕

𝜕𝑡

ˆx

𝑡

SSM



ˆx

𝑡





, (7)

and data loss

data

= MSE(ˆx

𝑡

data

). (8)

The mean squared error is denoted by MSE(). The

initial-condition loss, if sufﬁciently minimized, en-

sures that for 𝑡 = 0 the PINC predicts the initial state

. The physics loss L

phys

includes the system’s

dynamics equation in state-space form and therefore

governs learning of the physics model. The deriva-

tive of the predicted state

𝜕

𝜕𝑡

ˆx

𝑡

is calculated through

graph-based automatic differentiation with respect to

the time input 𝑡. In principle, these two losses are

sufﬁcient to learn the system dynamics. Optionally, a

data loss L

data

can be added, to include data sets ob-

tained from real hardware to incorporate effects that

may not be captured by the ﬁrst-principles model. As

these losses generally have different magnitudes and

convergence behaviors, they are weighted using the

factors 𝜆

𝑖

, which are either determined empirically or

by loss-weighting schemes as mentioned in Sec. 1.

In the training process, the loss functions L

and

phys

are evaluated on randomly sampled collocation

points in the intervals given in (3) and (4). One sam-

pling method commonly applied is Latin hypercube

sampling. Using backpropagation, the FNN parame-

ters θ are optimized using the total loss in (5).

After training, the PINC can be employed for dy-

namic state prediction in self loop as visualized in

Fig. 1(b) where 𝑘 indicates the current step. The step

size 𝑇 < 𝑇

is chosen constant for numerical integra-

tion at an operation frequency 𝑓 =

𝑇

3 DOMAIN-DECOUPLED PINN

We propose an alternative architecture to the PINC,

called the domain-decoupled physics-informed neural

network (DD-PINN) to address the limitations listed

in Sec. 1. The architecture of the DD-PINN in com-

parison to the PINC is visualized in Fig. 1(c–d) for

the training and prediction process, respectively. The

main modiﬁcation involves decoupling the time do-

main via

ˆx

𝑡

= g



,θ), 𝑡



= g(a,𝑡)+x

(9)

where f

now predicts the vector

a = f

,θ) (10)

which is used in an Ansatz function g. We restrict g

to the following two properties:

1. g is differentiable in closed form ∀𝑡 ∈ [0,𝑇

]

2. g(a, 0) ≡ 0

The ﬁrst property allows us to formulate the derivative

of the predicted state as

𝜕

𝜕𝑡

ˆx

𝑡



a,𝑡



. (11)

Therefore, we can calculate the gradients of the pre-

dicted state with respect to the time domain in analyt-

ical, closed form and do not require computationally

expensive automatic differentiation, addressing limi-

tation (I). Additionally, the initial condition is always

fulﬁlled, i.e.,

≡0. (12)

This addresses limitation (III) due to the second prop-

erty of g and introducing the initial state x

in (9).

The latter is comparable to a residual neural network

(ResNet) architecture (He et al., 2016). The loss-

balancing problem arising from (5) is then drastically

simpliﬁed. If no real data is used for training and only

a fast surrogate model is to be learned, no loss bal-

ancing is even necessary, as only one loss exists. The

consideration of lim. (II) can be found in section 3.2.

3.1 Formulating the Ansatz Function

The Ansatz function

g = (𝑔

, . . . , 𝑔

𝑗

, . . . , 𝑔

𝑚

)

⊤

(13)

consists of 𝑚 residual predictions 𝑔

𝑗

, where 𝑚 is the

number of states in the state-space model and 𝑔

𝑗

the

residual prediction of the 𝑗-th state variable Δˆ𝑥

𝑡, 𝑗

for

a given 𝑡. The residual state prediction is constructed

using 𝑛

𝑔

sub-functions.

𝑔

𝑗

𝑛

𝑔



𝑖=1

𝛼

𝑖 𝑗



𝜙

𝑔

(𝛽

𝑖 𝑗

𝑡 +𝛾

𝑖 𝑗

)−𝜙

𝑔

(𝛾

𝑖 𝑗

)



(14)

where 𝜙

𝑔

is a differentiable activation or base con-

struction function. We observe that the subtrahend

in (14) ensures that 𝑔

𝑗

(𝑡 = 0) ≡0, and the total length

of the vector

a = (α

⊤

,β

⊤

,γ

⊤

)

⊤

(15)

with coefﬁcient vectors α, β,γ results to 3·𝑚 ·𝑛

𝑔

. Fi-

nally, we obtain the derivative of the predicted state

¤𝑔

𝑗

𝑛

𝑔



𝑖=1

𝛼

𝑖 𝑗

𝛽

𝑖 𝑗

𝜙

𝑔

(𝛽

𝑖 𝑗

𝑡 +𝛾

𝑖 𝑗

) (16)

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

using the chain rule. Alternatively, a damping term

can be added to increase the numbers of parameters

in the sub-functions

𝑔

d𝑗

𝑛

𝑔



𝑖=1

𝛼

𝑖 𝑗



−𝛿

𝑖 𝑗

𝑡

𝜙

𝑔

(𝛽

𝑖 𝑗

𝑡 +𝛾

𝑖 𝑗

)−𝜙

𝑔

(𝛾

𝑖 𝑗

)



¤𝑔

d𝑗

𝑛

𝑔



𝑖=1

𝛼

𝑖 𝑗

−𝛿

𝑖 𝑗

𝑡



𝛽

𝑖 𝑗

𝜙

𝑔

(𝛽

𝑖 𝑗

𝑡 +𝛾

𝑖 𝑗

)

−𝛿

𝑖 𝑗

𝜙

𝑔

(𝛽

𝑖 𝑗

𝑡 +𝛾

𝑖 𝑗

)



, (17)

where now the length of vector a = (α

⊤

,β

⊤

,γ

⊤

,δ

⊤

)

⊤

results to 4·𝑚 ·𝑛

𝑔

The inclusion of the alternative damping term and

the choice of the base construction function 𝜙

𝑔

may

be decided based on the given state-space model char-

acteristics or empirical testing, which is similar to the

choice of a suitable activation functions of neural net-

works. In this study,

𝜙

𝑔

(𝑥) = sin(𝑥) and

𝜙

𝑔

(𝑥) = cos(𝑥) (18)

is used, which achieves good results on all three dif-

ferent systems in Sec. 4. However, any other differen-

tiable function may be used.

3.2 Higher-Order Excitation

In order to address limitation (II) concerning the zero-

order hold assumption, we introduce the option for

higher-order excitation input to the DD-PINN. As vi-

sualized in Fig. 1(c), the input to the FNN is the ex-

tended excitation vector u

∗

, that contains a concate-

nation of multiple excitation vectors inside the inter-

val [0,𝑇]. The function f

𝑢

calculates the interpolated

excitation vector

𝑡

= f

𝑢

∗

,𝑡) (19)

fed into the dynamics equation f

SSM

. In this study,

we limit the analysis to zero-, ﬁrst- and second-order

excitation, hereinafter referred to as the degree 𝛿

𝑢

u, using the following polynomial interpolations with

𝑡

∗

𝑡

𝑇

1. Zero-order hold, 𝛿

𝑢

= 0, similar to PINC:

∗

= u

, f

𝑢

= u

(20)

2. First order, 𝛿

𝑢

= 1:

∗

= (u

⊤

𝑇

)

⊤

, f

𝑢

= u

𝑇

𝑡

∗

+(1−𝑡

∗

(21)

3. Second order, 𝛿

𝑢

= 2:

∗

= (u

⊤

𝑇/2

⊤

𝑇

)

⊤

𝑢

= (2u

−4u

𝑇/2

+2u

𝑇

)𝑡

∗

+(−3u

+4u

𝑇/2

−1u

𝑇

)𝑡

∗

. (22)

Here, u

𝑇/2

= u(𝑡 =

𝑇

) and u

𝑇

= u(𝑡 = 𝑇) denote the

excitation at half and full step interval. The FNN then

implicitly learns the polynomial interpolation of the

components in u

∗

. It can be noted that instead of this

polynomial interpolation, a polynomial Taylor expan-

sion may also be used. If applied for model predictive

control, a DD-PINN trained for ﬁrst-order or second-

order input can still predict the system behavior under

zero-order-hold assumption with u

= u

𝑇/2

= u

𝑇

. De-

pending on the system‘s characteristics, an optimized

excitation vector for the (large) time steps within

MPC could be transformed into a ﬁrst/second-order

excitation to better approximate ﬁne changes within

small time steps outside the MPC loop.

3.3 Application of the DD-PINN to

Control Scenarios and Its

Generality

When the PINC is operated in self-loop mode for pre-

diction of the dynamical system behavior as visual-

ized in Fig. 1(b), only a relatively short time horizon

is predicted at once compared to applications of the

classical PINN. We therefore argue it is sufﬁciently

accurate to model this short interval with the Ansatz

function 𝑔 of the DD-PINN without the need of a high

number of construction functions 𝑛

𝑔

. The DD-PINN

also maintains the generality of the PINC, as the pa-

rameter 𝑛

𝑔

as well as the FNN size can be chosen

sufﬁciently large to model any arbitrary, continuous

function. It shall also be noted, that the concept pro-

posed here is not limited to the time domain but may

be applied to the spatial domain or others for the so-

lution of ODEs/PDEs.

4 VALIDATION

In this section, the PINC and the DD-PINN are eval-

uated on three dynamical systems visualized in Fig. 2

for comparison. The PINC and DD-PINN are both

trained and implemented using PyTorch. LRA is used

to balance the initial-condition loss (only PINC), the

data loss, and the physics loss. A data loss is only in-

cluded for the nonlinear mass-spring-damper system.

It is noted that these data points are not necessary for

training but demonstrate the capability of the PINC

and DD-PINN to include data that may be obtained

from a real system. For all systems, the Adam op-

timization algorithm is used with an initial learning

rate of 𝛼

init

= 0.001. A learning-rate scheduler that

reduces the learning rate on a plateau is employed

and the training is stopped early if the rate falls below

Domain-Decoupled Physics-informed Neural Networks with Closed-Form Gradients for Fast Model Learning of Dynamical Systems

(a)

(b)

(c)

𝑘, 𝑘

𝑑

𝑚

𝑘

𝑑

𝑚

𝑢

𝑞

−𝑢 𝑢

𝑚

𝑞

𝑚

𝑞

𝑚

𝑞

𝑚

𝑞

𝑚

𝑞

𝑔

𝑞

𝑢

𝑚

Figure 2: Three dynamical systems for evaluation: (a) nonlinear mass-spring-damper system, (b) ﬁve-mass-chain system, (c)

two-link manipulator system used in (Nicodemus et al., 2022).

a certain threshold of 𝛼

min

= 5 ·10

−8

. For the FNN

in all models, the Gaussian error linear unit (GELU)

activation function is used (Hendrycks and Gimpel,

2016). The ﬁrst system is trained on an Intel Cascade

Lake Xeon Gold 6230N CPU with 16 GB of RAM

and the latter two on an Intel Core i9-10900X CPU

with 64 GB of RAM. The training and neural network

parameters for each system are listed in Table 1 in the

appendix and an overview of the results are given in

Table 2–3.

4.1 Nonlinear Mass-Spring-Damper

System

The nonlinear mass-spring-damper system visualized

in Fig. 2(a) consists of a mass 𝑚 that is connected to a

ﬁxed wall through a spring with linear stiffness 𝑘 and

a cubic component of 𝑘

as well as a damper with

coefﬁcient 𝑑. An external force 𝑢 is applied to the

mass and the displacement 𝑞 and velocity ¤𝑞 denote

the system’s state x = (𝑞, ¤𝑞)

⊤

. The state-space model

results to

x =



¤𝑞

𝑢−𝑑 ¤𝑞−𝑘𝑞−𝑘

𝑞

𝑚



(23)

with 𝑚 = 0.001kg, 𝑑 = 0.001Ns/m, 𝑘 = 1N/m, and

𝑘

= 15N/m

Both architectures, the PINC and the DD-PINN,

are evaluated for three operation frequencies 𝑓 ∈

{50Hz,100Hz,200Hz}. The DD-PINN is further

evaluated in all combinations for zero-, ﬁrst-, and

second-order excitation 𝛿

𝑢

∈ {0,1,2} as described in

section 3.2 and for different numbers of base con-

struction functions 𝑛

𝑔

∈ {5,20, 50} – resulting in a

total of 27 DD-PINN models trained. For those mod-

els, a trigonometric base function 𝜙

𝑔

(𝑥) = sin(𝑥) is

used including the optional damping factor described

in (17).

For both architectures, an FNN with 96 neurons

and one hidden layer is chosen. The PINC is trained

for a maximum of 4,000 epochs on 20,000 colloca-

tion points, 40,000 initial condition points and 2,000

data points distributed over 400 batches. The DD-

PINN is trained for a maximum of 4,000 epochs as

0 2 4 6 8

−6

−3

𝑓 = 50 Hz

0 2 4 6 8

−7

−4

Validation error

𝑓 = 100 Hz

0 2 4 6 8 10

Training time in h

−8

−4

𝑓 = 200 Hz

PINC: L

phys

PINC: L

data

DD-PINN: L

phys

DD-PINN: L

data

Figure 3: Nonlinear mass-spring-damper system learning;

Reduction of validation errors over time. The DD-PINN

training time is signiﬁcantly lower.

well. The fast calculation of closed-form gradients al-

lows us to increase the count of collocation points to

375,000 distributed over 1,500 batches. Data points

are obtained from a simulation of the state-space

model (23).

The sampling ranges for the collocation points

(𝑞

min

, ¤𝑞

min

,𝑢

min

,𝑡

min

)

= (−0.4m,−18m/s,−1N,0s),

(𝑞

max

, ¤𝑞

max

,𝑢

max

,𝑡

max

)

= (0.4m,18m/s,1N,

1.1

𝑓

) (24)

are also used for normalization of the network inputs.

The training process is visualized in Fig. 3 show-

ing the validation physics loss L

phys

decrease over

time. We observe that the DD-PINN models converge

in average 8.7 times (minimum 4 .7, maximum 18.9)

faster than the PINC models despite a 9.375 times

higher count of collocation points per epoch. The

minimum validation error reached by the DD-PINN

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

−0.25

0.00

0.25

𝑞 in m

−10

¤𝑞 in m/s

0.0 0.5 1.0 1.5 2.0

time in s

𝑢 in N

Sim

PINC: 𝑓 = 100 Hz

DD-PINN: 𝑓 = 50 Hz,

𝛿

𝑢

= 2, 𝑛

𝑔

= 5

Figure 4: Nonlinear mass-spring-damper system self-loop

prediction of a test path for selected PINN and DD-PINN

models. Both models exhibit high prediction accuracy.

50 Hz

200 Hz

100 Hz

Figure 5: Parameter study of the nonlinear mass-spring-

damper system learning results; Scaled MSE to ground truth

of 1 s self-loop prediction path over relative prediction time.

DD-PINN models achieve signiﬁcantly higher prediction

accuracy for slightly slower prediction time.

networks is also lower in the case of 100 and 200 Hz,

but slightly higher for 50 Hz. It can also be seen that

the DD-PINN reaches a drastically lower validation

error for the data loss. The trained systems are evalu-

ated on a test path in self-loop prediction of which the

PINC model at 𝑓 = 100 Hz and a selected DD-PINN

model ( 𝑓 = 50Hz, 𝛿

𝑢

= 2, 𝑛

𝑔

= 5) are visualized in

Fig. 4. We observe an accurate and stable self-loop

prediction of both models on a time horizon of more

than 5 s.

A comparison of accuracy over self-loop predic-

tion time is given in Fig. 5. Here, the MSE between

the predicted path from Fig. 4 and the ground truth

is calculated until 𝑡 = 1 s and visualized over the pre-

diction time in relation to the PINC model at 𝑓 = 100

Hz. We observe that a higher frequency, i.e., a shorter

shooting interval generally corresponds to higher pre-

diction accuracy, coming at cost of prediction time.

Looking at each group of frequencies 50, 100, and

200Hz respectively, the DD-PINN achieves up to 2.1,

1.7, and 0.9 magnitudes higher accuracy with 𝛿

𝑢

> 0.

The prediction time is only slightly higher at approx.

1.2 times for the 100 Hz models and approx. 1.5 times

for the other two groups. A more detailed overview of

the results is also given in the appendix in Table 2.

4.2 Five-Mass-Chain System

The second system is a one-dimensional coupled

chain of ﬁve masses as visualized in Fig. 2(b). They

are connected at one side to a ﬁxed wall and be-

tween each other with a total of ﬁve stiffnesses 𝑘 and

dampers 𝑑. As an external input, the same load 𝑢 is

applied antagonistically at the ﬁrst and last mass. The

state vector is deﬁned in relative coordinates with dis-

placements and velocities

q = (𝑞

, 𝑞

)

⊤

q = ( ¤𝑞

, ¤𝑞

)

⊤

, (25)

resulting in a total of ten states in the state vector

x = (q

⊤

)

⊤

. (26)

The state vector is evaluated on a physics loss func-

tion based on the equation of motion

q(𝑡)+D

q(𝑡)+Kq(𝑡) = P 𝑢(𝑡) (27)

with mass matrix M , stiffness matrix K, damping

Matrix D, and vector

P = (−1,0, 0,0, 1)

⊤

. (28)

Two DD-PINN models are trained at 𝑓 = 50 Hz

with 𝛿

𝑢

∈ {0,2}. The hyper parameters are set to

𝑛

𝑔

= 20, 𝜙

𝑔

(𝑥) = sin(𝑥) and the optional damping

factor given in (17) is used. This system is hard to

predict accurately under the ZOH assumption show-

cased by one PINC model trained at 𝑓 = 50Hz. For

the DD-PINN and PINC, an FNN with 128 neurons

and two hidden layers is chosen. The PINC is trained

on 20,000 collocation points, 40, 000 initial condition

points over 400 batches and without data loss. The

DD-PINN is trained for 2,500 epochs, using 250,000

collocation points distributed over 500 batches. The

sampling ranges for the collocation points and input

normalization are determined based on the simulated

test path used, including a margin of 10%.

Fig. 6 shows the validation physics loss reduc-

tion during the training process. The DD-PINNs both

converged at around 4 h while the PINC training was

Domain-Decoupled Physics-informed Neural Networks with Closed-Form Gradients for Fast Model Learning of Dynamical Systems

0 50 100 150

Training time in h

−1

−2

−3

−4

Validation error

Training stopped

PINC: L

phys

PINC: L

DD-PINN: L

phys

Figure 6: Five-mass-chain system learning: Reduction of

validation errors over training time. DD-PINN converged

within 5 h, while the PINC training was stopped after 145 h.

0.0000

0.0005

0.0010

Displacements 𝑞

𝑖

𝑞

−0.02

0.00

0.02

Velocities ¤𝑞

𝑖

¤𝑞

0.0 0.2 0.4 0.6 0.8 1.0

time in s

Exc. 𝑢

Sim

PINC (until 𝑡 = 0.4)

DD-PINN: 𝛿

𝑢

= 0

(until 𝑡 = 0.4)

DD-PINN: 𝛿

𝑢

= 2

Figure 7: DD-PINN self-loop prediction of a test path for

the ﬁve-mass-chain system including poor predictions for

models with ZOH assumption until 𝑡 = 0.4s . The DD-

PINN’s higher order excitation enables smooth prediction.

stopped after 145h and 2,400 epochs of training. As

we can observe in Fig. 7, the ZOH assumption is

not sufﬁcent for a prediction frequency at 50 Hz as

strong oscillations are visible in both, the PINC’s and

the DD-PINN’s self-loop prediction (only plotted un-

til 𝑡 = 0.4s). This is due to the interval-wise constant

input acting as a step excitation that excites higher

modes of the system.

It can be seen that having an interval-wise

quadratic excitation input to the DD-PINN, the pre-

dicted path becomes stable and accurate to the simu-

lated system response. The self-loop prediction time

for the DD-PINN models was measured to be 33%

and 34% slower than the PINC for 𝛿

𝑢

= 0 and 𝛿

𝑢

= 2,

respectively, noting that the increased order in excita-

tion does not come along with an increase in predic-

tion time for the DD-PINN.

4.3 Two-Link Manipulator System

The two-link manipulator system visualized in

Fig. 2(c) denotes a Schunk PowerCube robot with

dynamics identiﬁed in (Fehr et al., 2022). In previ-

ous work (Nicodemus et al., 2022), a PINC was used

in MPC for fast and accurate prediction of the sys-

tem dynamics. The manipulator consists of one ﬁxed

link in upright position following two sequential links

connected by two joints at angle 𝑞

and 𝑞

at which

the motor currents 𝑢

, 𝑢

apply. With the general-

ized coordinates q = (𝑞

, 𝑞

)

⊤

and the state vector

x = (q

⊤

)

⊤

, the state-space model results to

x =



M (q)

−1



h(q,

q)−k(q,

q)+Bu





(29)

with mass matrix M , input matrix B (including mo-

tor constant), vector of centrifugal and Coriolis forces

k and vector of gravitational and damping forces h

adopted from (Nicodemus et al., 2022).

One PINC and one DD-PINN model at 𝑓 = 5 Hz

are trained. The DD-PINN is set to 𝛿

𝑢

= 0 for direct

comparison, using 𝑛

𝑔

= 20, 𝜙

𝑔

(𝑥)= sin(𝑥) and the op-

tional damping factor given in (17). For the PINC, an

FNN with 64 neurons and four hidden layers (equiva-

lent to (Nicodemus et al., 2022)) is chosen and 128

neurons with two hidden layers for the DD-PINN.

The PINC is trained for 7,000 epochs on 20,000 col-

location points, 40,000 initial condition points over

400 batches and without data loss. The DD-PINN

is trained for 2,500 epochs, using an increased count

of collocation points of 1 ·10

distributed over 1,000

batches. The used sampling ranges for the collocation

points and input normalization are

(𝑞

1,min

, 𝑞

2,min

, ¤𝑞

1,min

, ¤𝑞

2,min

,𝑢

1,min

,𝑢

2,min

,𝑡

min

)

= (−𝜋, −𝜋, −1s

−1

,−1s

−1

,−0.6A, −0.6A, 0s),

(𝑞

1,max

, 𝑞

2,max

, ¤𝑞

1,max

, ¤𝑞

2,max

,𝑢

1,max

,𝑢

2,max

,𝑡

max

)

= (𝜋, 𝜋,1s

−1

,1s

−1

,0.6A, 0.6A,

1.1

𝑓

). (30)

Fig. 9 shows the validation physics loss reduction

during the training process. The training of the PINC

did not converge within 7000 epochs and was discon-

tinued due to the exceedingly high training time of

over 210 h. The DD-PINN converged in a compar-

atively short time of 7.4 h while reaching a signiﬁ-

cantly lower validation error. This difference shows

effect in the self-loop prediction on a custom test path

in Fig. 8. Here, the PINC’s prediction diverges after

7 s. This is comparable to previous work where the

PINC prediction diverged in self-loop after approx.

3 s for the same system (Nicodemus et al., 2022). In

contrast, the prediction of the DD-PINN remains sta-

ble and accurate for the whole 28s-long test path. For

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

−1

𝑞

in rad

Sim

DD-PINN,

𝑓 = 5 Hz, 𝛿

𝑢

= 0

PINC, 𝑓 = 5 Hz

𝑞

in rad

−0.5

0.0

0.5

¤𝑞

in rad/s

−0.5

0.0

0.5

¤𝑞

in rad/s

0 5 10 15 20 25

time in s

0.0

0.5

𝑢

in A

0 5 10 15 20 25

time in s

−0.25

0.00

0.25

𝑢

in A

Figure 8: PINC and DD-PINN self-loop prediction of a test path for the two-link manipulator system. The DD-PINN’s

prediction is stable while the PINC’s prediction diverges.

0 50 100 150 200

Training time in h

−1

−2

−3

−4

−5

Validation error

Training stopped

PINC: L

phys

PINC: L

DD-PINN: L

phys

Figure 9: Two-link manipulator system learning: Reduc-

tion of validation errors over training time. DD-PINN con-

verged within 7.4h, while the PINC training was stopped

after more than 210h.

this system, the self-loop prediction time for the DD-

PINN model was measured to be approximately equal

to the computation time of the PINC. The results of

the two-link manipulator system as well as the ﬁve-

mass-chain system are also given in the appendix in

Table 3.

4.4 Discussion

The results show that the DD-PINN successfully ad-

dresses the three limitations of the PINC as formu-

lated in Sec. 1: (I) Without the need of automatic

differentiation to calculate the physics loss, training

times were reduced signiﬁcantly for all three tested

systems by a factor of 5–38 while evaluating 10–25

times more collocation points per epoch. A higher

count of collocation points improves the coverage

over the sampling ranges. In the majority of cases,

the DD-PINN also reached lower physics validation

errors with a signiﬁcant improvement for the two-

link manipulator system that enabled stable self-loop

prediction in contrast to the PINC. (II) The limi-

tations of the zero-order hold assumption are over-

come by introducing the option for higher-order ex-

citation. This was shown to be necessary for the

ﬁve-mass-chain system to prevent induced vibrations

from the interval-wise step excitation for 𝛿

𝑢

= 0.

Also, a higher-order excitation in combination with

a longer shooting interval enables faster prediction

while being more accurate as shown for the nonlinear

mass-spring damper system. Lastly (III), the initial-

condition loss for the DD-PINN is negligible (L

≡

0). While a loss-balancing scheme such as LRA may

still be employed when the data loss is used, the bal-

ancing process is strongly simpliﬁed. This is because

the reduction of the physics loss towards a solution of

the initial-value problem for the PINC depends on a

sufﬁciently low initial-condition loss.

The DD-PINN therefore provides a solution to

these limitations while being easy to implement and

interchangeable to the classical PINC architecture.

Also, it is compatible with a wide range of arbitrary

Ansatz functions that may be chosen suitable for the

problem at hand.

5 CONCLUSIONS

Physics-informed neural networks (PINNs) represent

a well-established machine-learning framework that

combines learning the solution of differential equa-

tions with supervised learning based on data. In its

adaptation to dynamical systems, namely the PINN

Domain-Decoupled Physics-informed Neural Networks with Closed-Form Gradients for Fast Model Learning of Dynamical Systems

for control (PINC), we identiﬁed three primary limi-

tations: (I) Long training times for large and/or com-

plex state-space models, (II) limitations arising from

the zero-order-hold assumption for excitation, and

(III) the need for hyperparameter-sensitive loss bal-

ancing schemes.

In this study, we introduced the domain-decoupled

physics-informed neural network (DD-PINN) as a so-

lution to these limitations. We ﬁrst formulated the

DD-PINN architecture, showcasing how it enables

calculation of gradients for the physics loss in closed

form, is compatible to higher-order excitation input,

and always has an initial-condition loss of zero. We

then compared the DD-PINN to the PINC in simula-

tion for three benchmark systems. The results demon-

strated that the DD-PINN signiﬁcantly reduces train-

ing times while maintaining or surpassing the predic-

tion accuracy of the PINC. Thereby, the self-loop pre-

diction time of the DD-PINN is comparable to the

PINC.

The DD-PINN allows for fast and accurate learn-

ing of large and complex dynamical systems, which

were previously out of reach for the PINC. Its fast

prediction abilities create opportunities for enabling

MPC in larger dynamical systems, where traditional

methods like numerical integrators are too slow, and

training a PINC is not practical. Here, the data

efﬁciency of physics-informed machine learning re-

mains, making it possible to integrate sparse datasets

with the system-governing physical equations. Fu-

ture work could explore using DD-PINN for higher-

dimensional nonlinear systems to realize accurate

state estimation or model predictive control.

ACKNOWLEDGEMENTS

This work was partially funded by the Ger-

man Research Foundation (DFG, project numbers

405032969 and 433586601) and the Lower Saxo-

nian Ministry of Science and Culture in the program

zukunft.niedersachsen.

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Domain-Decoupled Physics-informed Neural Networks with Closed-Form Gradients for Fast Model Learning of Dynamical Systems

APPENDIX

Table 1: PINC and DD-PINN training and neural-network parameters for the three evaluated systems.

Nonlinear mass-spring-damper Five-mass chain Two-link manipulator

PINC DD-PINN PINC DD-PINN PINC DD-PINN

Epochs ≤ 4,000 ≤ 4,000 2,400 2,500 7,000 2,500

Batches 400 1,500 400 500 400 1,000

𝑛

40,000 0 40,000 0 40,000 0

𝑛

collo

20,000 375,000 20,000 250,000 20,000 1·10

𝑛

data

2,000 2,000 0 0 0 0

Neurons 96 96 128 128 64 128

Hidden layers 1 1 2 2 4 2

Table 2: PINC and DD-PINN training results for the nonlinear mass-spring-damper. Best values are highlighted in bold.

Nonlinear mass-spring-damper

PINC DD-PINN

Number of

models trained

1 1 1 9 9 9

𝑓 in Hz 50 100 200 50 100 200

Training

time in h

8.04 9.22 9.70

0.64

1.65

0.76

1.70

0.54

1.61

5.20·10

−7

1.67·10

−7

1.11·10

−7

phys

2.26·10

−5

2.05·10

−6

3.72·10

−7

2.90·10

−5

7.67·10

−5

1.08·10

−6

4.87·10

−6

5.49 ·10

−8

2.42·10

−7

data

1.38·10

−4

1.54·10

−5

4.55·10

−7

1.21·10

−8

1.78·10

−4

6.79·10

−10

1.99·10

−5

1.15 ·10

−11

7.83·10

−8

Scaled MSE

on test path

1.93·10

−1

1.92·10

−2

1.92·10

−3

1.53·10

−3

2.40·10

−1

4.30·10

−4

3.14·10

−2

2.26 ·10

−4

4.07·10

−3

Relative pred.

time

0.43 1.00 1.62

0.57

0.67

1.15

1.26

2.34

2.79

Values denote min and max values among the different conﬁgurations with varying degree of excitation 𝛿

𝑢

∈ {0,1, 2}

and different numbers of base construction functions 𝑛

𝑔

∈ {5,20, 50}.

Table 3: PINC and DD-PINN training results for the ﬁve-mass-chain and two-link manipulator systems. Best values per

system are highlighted in bold.

Five-mass chain Two-link manipulator

PINC DD-PINN

Training

time in h

144.5

4.29

3.75

211.0 7.41

8.80·10

−5

8.90·10

−6

phys

6.27 ·10

−3

8.43·10

−3

8.22·10

−3

2.46·10

−2

3.22 ·10

−3

data

Scaled MSE

on test path

1.78·10

−1

2.42·10

−1

1.27 ·10

−3

9.17·10

−2

1.68 ·10

−3

Relative pred. time 1.00

1.33

1.34

1.00 1.03

Values for the DD-PINN for the ﬁve-mass-chain system correspond to the conﬁgurations with 𝛿

𝑢

= 0 and 𝛿

𝑢

= 2.

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