On the Forward Invariance of Neural ODEs

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Wei Xiao 1Tsun-Hsuan Wang 1Ramin Hasani 1Mathias Lechner 1

Yutong Ban 1Chuang Gan 2Daniela Rus 1

Abstract

We propose a new method to ensure neural ordi-

nary differential equations (ODEs) satisfy output

speciﬁcations by using invariance set propaga-

tion. Our approach uses a class of control barrier

functions to transform output speciﬁcations into

constraints on the parameters and inputs of the

learning system. This setup allows us to achieve

output speciﬁcation guarantees simply by chang-

ing the constrained parameters/inputs both dur-

ing training and inference. Moreover, we demon-

strate that our invariance set propagation through

data-controlled neural ODEs not only maintains

generalization performance but also creates an ad-

ditional degree of robustness by enabling causal

manipulation of the system’s parameters/inputs.

We test our method on a series of representation

learning tasks, including modeling physical dy-

namics and convexity portraits, as well as safe

collision avoidance for autonomous vehicles.

1. Introduction

Neural ODEs (Chen et al.,2018) are continuous deep learn-

ing models that enable a range of useful properties such as

exploiting dynamical systems as an effective learning class

(Haber & Ruthotto,2017;Gu et al.,2021), efﬁcient time

series modeling (Rubanova et al.,2019;Lechner & Hasani,

2022), and tractable generative modeling (Grathwohl et al.,

2018;Liebenwein et al.,2021).

Neural ODEs are typically trained via empirical risk min-

imization (Rumelhart et al.,1986;Pontryagin,2018) en-

dowed with proper regularization schemes (Massaroli et al.,

2020) without much control over the behavior of the ob-

tained network and over the ability to account for coun-

Computer Science and Artiﬁcial Intelligence Lab, Mas-

sachusetts Institute of Technology, Cambridge, MA, USA.

MIT-

IBM Watson AI Lab. Videos and code are available on the

website:

https://weixy21.github.io/invariance/

Correspondence to: Wei Xiao <weixy@mit.edu>.

Proceedings of the

40 th

International Conference on Machine

Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright

2023 by the author(s).

Input Output

Input

Invariance

Neural ODE

𝐼(𝑡)

Invariance propagation

Output

spec

What if there is an

obstacle

In the flow of a neural ODE?

Neural ODE with no Invariance

Neural ODE + Invariance

We use Neural ODEs with Invariance

Figure 1.

Invariance Propagation for neural ODEs. Output speciﬁ-

cations can be guaranteed with invariance, including speciﬁcation

satisfaction between samplings, e.g., spiral curve regression with

critical region avoidance.

terfactual inputs (Vorbach et al.,2021). For example, a

well-trained neural ODE instance that learned to chase a

spiral dynamic (Fig. 1B), would not be able to avoid an

object on its ﬂow, even if it has seen this type of output

speciﬁcation/constraint during training. This shortcoming

demands a fundamental ﬁx to ensure the safe operation of

these models speciﬁcally in safety-critical applications such

as robust and trustworthy policy learning, safe robot control,

and system veriﬁcation (Lechner et al.,2020;Kim et al.,

2021;Hasani et al.,2022).

In this paper, we set out to ensure neural ODEs satisfy

output speciﬁcations. To this end, we introduce the concept

of propagating invariance sets. An invariance set is a form

of speciﬁcation consisting of physical laws, mathematical

expressions, safety constraints, and other prior knowledge of

the structure of the learning task. We can ensure that neural

ODEs are invariant to noise and afﬁne transformations such

as rotating, translating, or scaling an input, as well as to

other uncertainties in training and inference.

To propagate invariance sets through neural ODEs we can

use Lyapunov-based methods with forward invariance prop-

erties such as a class of control barrier functions (CBFs)

(Ames et al.,2017), to formally guarantee that the output

arXiv:2210.04763v2 [cs.LG] 31 May 2023

On the Forward Invariance of Neural ODEs

speciﬁcations are ensured. In order to account for non-

linearity of the model, high-order CBFs (Xiao & Belta,

2019), a general form of CBFs, are required since high-

relative degree constraints are introduced in such cases.

CBFs perform this via migrating output speciﬁcations to

the learning system’s parameters or its inputs such that we

can solve the constraints via forward calls to the learning

system equipped with a quadratic program (QP). However,

doing this requires a series of non-trivial novelties which we

address in this paper. 1. CBFs are model-based Lyapunov

methods, thus, they can only be used with data and systems

with known dynamics. Here, we extend their formalism

to work with unknown dynamics by the properties of neu-

ral ODEs. 2. CBFs are typically applied to systems with

afﬁne transformations. For neural ODEs with nonlinear

activations, the propagation of invariance sets becomes a

challenge. We ﬁx this by incorporating a virtually linear

space within the neural ODE to ﬁnd simple parameter/input

constraints.

Going back to Fig. 1C, we observe that by applying our

forward invariance propagation method, we can correct the

model and force the system trajectories to stay away from

the red obstacles while maintaining the path of the ground

truth spiral curve.

In summary, we make the following new contributions:

•

We incorporate formal guarantees into neural ODEs via

invariance set propagation.

•

We use the class of Higher-order CBFs (Xiao & Belta,

2022) to propagate the invariance set in neural ODEs while

addressing their challenges such as handling unknown

dynamics and the nonlinearity of the neural ODEs as well

as connecting the order concept in HOCBFs to that of

network depth in neural ODEs.

•

We demonstrate the effectiveness of our method on a vari-

ety of learning tasks and output speciﬁcations, including

the modeling of physical systems and the safety of neural

controllers for autonomous vehicles.

2. Preliminaries

In this section, we provide background on neural ODEs and

forward invariance in control theory.

2.1. Neural ODEs

A neural ordinary differential equation (ODE) is deﬁned in

the form (Chen et al.,2018):

x(t) = fθ(x(t)),(1)

where

n∈N

is state dimension,

x∈Rn

is the state and

denotes the time derivative of

fθ:Rn→Rn

is a

neural network model parameterized by

. The output of

the neural ODE is the integral solution of (1). It can also

take in external input, where the model is deﬁned as:

x(t) = f′

θ(x(t),I(t)),(2)

where

nI∈N

is external input dimension,

I(t)∈RnI

f′

θ:

Rn×RnI→Rn

is a neural network model parameterized

by θ. For notation convenience, we write as,

fθ=fθK,K+1 ◦ · · · ◦ fθ1,2(3)

where

is the number of layers,

fθk,k+1 , k ∈[1, K]

is the forward process of the

’th layer, and we denote

zk= (fθk,k+1 ◦ · · · ◦ fθ1,2)(·)∈Rnk

the intermediate repre-

sentation at the

’th layer and

zK=˙

is the output.

nk∈N

denotes the number of neurons at layer kand nK=n.

2.2. Forward Invariance in Control Theory

Consider an afﬁne control system of the form

x=f(x) + g(x)u(4)

where

x∈Rn

f:Rn→Rn

and

g:Rn→Rn×q

are

locally Lipschitz, and

u∈U⊂Rq

, where

denotes a

control constraint set.

Deﬁnition 2.1. (Set invariance): A set

C⊂Rn

is forward

invariant for system (4) if its solutions for some

u∈U

starting at any x(t0)∈Csatisfy x(t)∈C, ∀t≥t0.

Deﬁnition 2.2. (Relative degree): The relative degree of

a differentiable function

b:Rn→R

(or constraint

b(x)≥

) with respect to the system (4) is the number of times

b(x)

needs to be differentiated along dynamics (4) until

any component of

explicitly shows in the corresponding

derivative.

Deﬁnition 2.3. (Class

function): A Lipschitz continuous

function

α: [0, a)→[0,∞), a > 0

belongs to class

if it

is strictly increasing and α(0) = 0.

Deﬁnition 2.4. (High Order Barrier Function (HOBF)):

A function

b:Rn→R

of relative degree

is a HOBF with

a sequence of functions

ψi:Rn→R

such that

ψm(x)≥0

ψi(x) := ˙

ψi−1(x) + αi(ψi−1(x)), i ∈ {1, . . . , m},

(5)

where

ψ0(x) := b(x)

and

αi(·)

is a

(m−i)

’th order differ-

entiable class Kfunction. We deﬁne a sequence of sets,

Ci:= {x∈Rn:ψi−1(x)≥0}, i ∈ {1, . . . , m}.(6)

Deﬁnition 2.5. (High Order Control Barrier Function

(HOCBF) (Xiao & Belta,2022)): Let

ψi

and

be deﬁned

(5)

and

(6)

, respectively, for

i∈ {1, . . . , m}

. A function

b:Rn→R

is a HOCBF of relative degree

if there exists

On the Forward Invariance of Neural ODEs

(m−i)th

order differentiable class

functions

αi, i ∈

{1, . . . , m}such that

sup

u∈U

[Lm

fb(x)+[LgLm−1

fb(x)]u+O(b(x))

+αm(ψm−1(x))] ≥0,

(7)

for all

x∈C1∩,...,∩Cm

and

denote Lie deriva-

tives w.r.t.

along

and

, respectively, and

O(b(x)) =

Pm−1

i=1 Li

f(αm−i◦ψm−i−1)(x)

. The satisfaction of (7) is

equivalent to the satisfaction of ψm(x)≥0deﬁned in (5).

The HOCBF is a general form of the CBF (Ames et al.,

2017) (a HOCBF with

m= 1

degenerates to a CBF), and

it can be applied to arbitrary relative degree systems, such

as the invariance propagation to nonlinear layers of a neural

ODE in this work.

Theorem 2.6. (Xiao & Belta,2022)): Given a HOCBF

b(x)

from Def. 2.5 with the sets

C1, . . . , Cm

deﬁned by (6),

x(t0)∈C1∩,...,∩Cm

, then any Lipschitz continuous

controller

u(t)

that satisﬁes the constraint in (7),

∀t≥t0

renders C1∩,...,∩Cmforward invariant for system (4).

In this work, we map the forward invariance in control

theory to forward invariance in neural ODEs, where we

tackle arbitrary dynamics deﬁned by neural ODE

fθ

(which

can be nonlinear as opposed to an afﬁne control system in

the form of (4)).

3. Invariance Propagation

In this section, we present the theoretical framework of In-

variance Propagation (IP) to guarantee forward invariance

(in short, invariance) of a neural ODE. We ﬁrst provide

formalisms of the proposed method. Then, we describe

invariance propagation to (i) linear layer (ii) nonlinear layer

(iii) external input.

Output Speciﬁcation. A continuously differentiable func-

tion

h:Rn→R

constructs an output speciﬁcation

h(x)≥0

for a neural ODE. Typical output speciﬁcations in-

clude system safety (e.g., collision avoidance in autonomous

driving), physical laws (e.g., energy conservation), mathe-

matical formulae (e.g., Cauchy Schwarz inequality), etc.

Deﬁnition 3.1. (Forward Invariance in Neural ODE):

The (forward) invariance of a neural ODE

(1)

(2)

with

fθ

is deﬁned w.r.t. its output speciﬁcation

h(x)≥0

such

that if

h(x(t0)) ≥0

, then

h(x(t)) ≥0,∀t≥t0

, where

x(t) = Rt

t0fθ(τ)dτ

. Intuitively, this property guarantees

the satisfaction of output speciﬁcation is forwarded in the

neural ODE across time.

Deﬁnition 3.2. (Invariance Propagation (IP)): Given an

output speciﬁcation

h(x)≥0

and a neural ODE

fθ

, invari-

ance propagation describes a procedure to ﬁnd a constraint

Ψ(d)≥0

, where

Ψ : Rnd→R

and

(

nd∈N

is its dimen-

sion) is either (i) a subset of parameters

, (ii) the external

input

, or (iii) other auxiliary variables for the neural ODE,

such that if

Ψ(d)≥0

, forward invariance deﬁned in Def.

3.1 is satisﬁed. Intuitively, IP casts invariance w.r.t. output

speciﬁcation to pose constraints on non-output in ODEs.

3.1. Invariance Propagation to Linear Layers

We start with a simple case where invariance is propagated

to linear layers of the neural ODE, which normally occurs

at the output layer without nonlinear activation functions.

Neural ODE Reformulation. Without loss of generality,

we follow (3) and assume a linear output layer fθK−1,K ,

i=1

θi

K−1,K zi

K−1=θP

K−1,K zP

K−1+θN

K−1,K zN

K−1

(8)

where

zK−1= (fθK−1,K−2◦ · · · ◦ fθ1,2)(x)

θi

K−1,K

is the

’th column of

θK−1,K ∈Rn×nK−1

K−1

is the

’th entry

zK−1∈RnK−1

and

describe sets of columns that

are updatable parameters (that the invariance is propagated

to) and constants, respectively. We drop the bias term for

cleaner notation.

Propagation to Linear Layers. Our goal is to propagate the

invariance to a subset of parameters. We treat

θP

K−1,K

as a

variable while taking other parameters

θN

K−1,K

as constants.

Given an arbitrary output speciﬁcation

h(x)≥0

, we can

deﬁne a ψ1function in the form:

ψ1(x, θP

K−1,K ) := dh(x)

dxfθ(x) + α1(h(x)),(9)

where

α1(·)

is a class

function. Note that

θP

K−1,K

is im-

plicitly deﬁned in

fθ

. Combining

(9)

with

(8)

, the following

theorem shows the invariance of the neural ODE (1):

Theorem 3.3. Given a neural ODE as in (8) and an output

speciﬁcation

h(x)≥0

, if there exist a class

function

α1

and θP

K−1,K such that with ψ1as in (9),

Ψ(θP

K−1,K |x) = ψ1(x, θP

K−1,K )≥0,(10)

for all

such that

h(x)≥0

, where

Ψ(θP

K−1,K |x) =

dxθP

K−1,K zP

K−1+dh

dxθN

K−1,K zN

K−1+α1(h(x))

, then the

neural ODE is forward invariant.

The proof and existence of α1are shown in Appendix A.1.

Brief Summary. Thm. 3.3 provides a condition on the

parameter

θP

K−1,K

that implies the invariance of the neural

ODE. In other words, by modifying the parameter θP

K−1,K

such that (10) is always satisﬁed, we can guarantee the

invariance. The algorithm is shown in the next section.

Moreover, since we only need to take the derivative of

h(x)

On the Forward Invariance of Neural ODEs

Neural ODE

   

Output invariance

(spec.)

 



     

Hidden invariance

 



   

Auxiliary

input invariance

Layer K



Invariance Propagation

Outputs



Inputs

Layer K-1

……

Layer +1

……

Layer 

Layer 

















……

Chain rule

Virtual

System

Figure 2.

Invariance propagation to an arbitrary layer of the neural

ODE with auxiliary virtual linear space.

once, as shown in (10), this is analogous to a ﬁrst-order

HOCBF (i.e., m= 1 in Def. 2.5).

IP to the Proper Parameters. By using the Theorem 3.3 ,

we can propagate the invariance to neural ODE parameters

in linear layers. However, note that we need to choose the

parameters such that all the output of the neural ODE are

able to be changed by modifying the target parameters. Oth-

erwise, the output speciﬁcation may fail to be guaranteed.

3.2. Invariance Propagation to Nonlinear Layers

In this section, we consider how we may efﬁciently propa-

gate the invariance to the weight parameters of an arbitrary

layer of the neural ODE (including the output layer with

nonlinear activation functions). Theoretically, we can prop-

agate the invariance to arbitrary layers using the existing

HOCBF theory. However, the resulting invariance enforce-

ment would be nonlinear programs (i.e., the HOCBF con-

straint (7) will be nonlinear in

), which are computation-

ally hard and inefﬁcient to solve. Moreover, the formulation

above does not allow us to incorporate the IP in the training

loop to address the conservativeness of the invariance as

discussed next. Our method works for both (1) and (2), so

we only consider (1) for simplicity.

An Auxiliary Linear System. Given a neural ODE

(1)

, we

want to propagate the invariance to the partial parameter

(similar to

(8)

) at the

’th layer

θP

k,k+1 ∈RnP

k+1×nk

with

k∈ {1, . . . , K}

and

k+1 ≤nk+1

. Then, we ﬂatten the

matrix parameter

θP

k,k+1

to a vector form

θP

k∈RdP

row-

wise, where

k=nP

k+1nk

is the dimension of the vector.

Instead of directly propagating the invariance to the parame-

ter

θP

and resulting nonlinear constraints, we propagate the

invariance to an auxiliary linear system:

θP

k=AP

kθP

k+BP

kuP

k(11)

where

k∈RdP

k×dP

k, BP

k∈RdP

k×dP

are chosen such

that the auxiliary system is controllable and

k∈RdP

the auxiliary control input. The exact choice of

and

may slightly impact the performance, which is further

discussed in Appendix B. This speciﬁc formulation allows

performing IP on

linearly (which will become clearer

later on) as opposed to directly on

θP

, which is susceptible

to nonlinearity. An overview is illustrated in Fig. 2.

Propagation to Auxiliary System. We ﬁrst propagate the

invariance to parameter

θP

by deﬁning a function

ψ1

similar

to (9), which is illustrated by the blue boxes in Fig. 2. Then,

we further propagate the invariance to the

in system (11)

by deﬁning another function ψ2(the red box in Fig. 2):

ψ1(x, θP

k) := dh(x)

dxfθ(x) + α1(h(x)),

ψ2(x, uP

k) := ∂ψ1

∂xfθ(x) + ∂ψ1

∂θP

θP

k+α2(ψ1(x, θP

k)),

(12)

where

α1(·), α2(·)

are class

functions and

ψ1, ψ2

are

deﬁned in the similar spirit to

(5)

. Remark that different

from

(9)

, here,

ψ1

is nonlinear regarding

θP

yet the newly

introduced

ψ2

is linear to the auxiliary variable

thanks

θP

depicting a linear system w.r.t.

as shown in

(11)

Combining (12) with (11), the following theorem shows the

invariance:

Theorem 3.4. Given a neural ODE deﬁned by (1) and

an output speciﬁcation

h(x)≥0

, if there exist class

functions α1, α2and uP

ksuch that with ψ1, ψ2as in (12),

Ψ(uP

k|x) = ψ2(x, uP

k)≥0,(13)

for all

that satisﬁes

h(x)≥0

and

ψ1(x)≥0

, where

Ψ(uP

k|x) = d2h(x)

dx2f2

θ(x)+ dh(x)

∂fθ(x)

∂θP

(AP

kθP

k+BP

kuP

k)+

(dh(x)

∂fθ(x)

∂x+dα1(h(x))

dx)fθ(x)+α2(ψ1(x))

, then the neu-

ral ODE is forward invariant.

The proof and existence of

α1, α2

are shown in Appendix

A.2. Intuitively, invariance is ﬁrst propagated via

ψ1

θP

then via

ψ2

, rendering a linear constraint in

(13)

. We

will further show how this enforces invariance in Sec. 4.2.

Brief Summary. Note that (13) is linear in

with the

assistance of system (11). Instead of directly changing the

parameters of the neural ODE for the invariance as Sec. 3.1,

we ﬁnd auxiliary control

that satisﬁes the constraint (13)

to dynamically change the parameters. Also, since we take

the derivative of

h(x)

twice, as in (13), this is analogous to

a second-order HOCBF (i.e., m= 2 in Def. 2.5)

IP to the Proper Parameters. While Thm. 3.4 allows us to

propagate the invariance to arbitrary neural ODE parameters,

the choice of the parameters may affect the performance,

e.g., the model’s accuracy). The speciﬁc parameter choice

depends on the model structure and the task’s output speciﬁ-

cation. In most cases, we may wish to choose the parameters

On the Forward Invariance of Neural ODEs

of the same layer to propagate the invariance to. However, it

is also possible to choose parameters of different layers, as

long as we deﬁne auxiliary dynamics for all the parameters

as in (11). The proposed method still works in such cases.

We may need to choose the parameters such that the output

of the neural ODE can all be changed, as in the linear case.

3.3. Invariance Propagation to External Input

We consider a neural ODE in the form of

(2)

with an external

input I.

Approach 1: As in Sec. 3.1, we may directly reformulate

(2) in the following afﬁne form:

x=fθ(x) + gθ(x)I,(14)

where

fθ

is deﬁned as in (1),

gθ:Rn→Rn×nI

is another

neural network parameterized by

. Then, we can use the

similar technique as in Sec. 3.1 to propagate the invariance

to the external input Ias they are both in afﬁne forms.

Approach 2: If we do wish to keep neural ODEs with

external input as in the form of (2), then we may deﬁne

auxiliary linear dynamics as in Sec. 3.2, and augment (2) by

the following form:

x=f′

θ(x,y),˙

y=Ay+BI,(15)

where

y∈RnI

is the auxiliary variable,

A∈RnI×nI, B ∈

RnI×nI

are deﬁned such that the linear system is controllable

(similar to

(11)

). Then, we can use the similar technique

as in Sec. 3.2 to propagate the invariance to the external

input

via the auxiliary variable

. In fact, the above neural

ODE becomes a stacked neural ODE, which will be further

studied (as discussed in Appendix E).

4. Enforcing Invariance in Neural ODEs

Here, we show how we proceed from the theoretical frame-

work in Sec. 3to efﬁcient algorithms of IP on neural ODEs.

4.1. Algorithms for Linear layers

Enforcing invariance. Enforcing the invariance of a neural

ODE is equivalent to the satisfaction of the condition in

Thm. 3.3. Also by proof in Appendix A.1, we can always

ﬁnd a class

function

α1(·)

such that there exists

θP

K−1,K

that makes (10) satisﬁed if

h(x(t0)) ≥0

. If

h(x1(t0)) ≤0

then the output of the neural ODE will be driven to satisfy

h(x)≥0

when the constraint (10) in Thm. 3.3 is satis-

ﬁed due to its Lyapunov property (Ames et al.,2012). The

enforcing of the invariance could vary in different applica-

tions and we do not restrict to exact methods. We provide a

minimum-deviation quadratic program (QP) approach.

Let

θP†

K−1,K ∈Rn×nP

K−1

denote the value of

θP

K−1,K

dur-

ing training or after training. Then, we can formulate the

following optimization:

θP∗

K−1,K = arg min

θP

K−1,K

||θP

K,K−1−θP†

K−1,K ||2,s.t. (10),

(16)

where

|| · ||

denotes the Euclidean norm. The above opti-

mization becomes a QP with all other variables ﬁxed except

θP

K−1,K

. This solving method has been shown to work in

(Ames et al.,2017) (Glotfelter et al.,2017) (Xiao & Belta,

2022). At each discretization step, we solve the above QP

and get

θP∗

K−1,K

. Then we set

θP

K−1,K =θP∗

K−1,K

during

the inference of the neural ODE. This way, we can enforce

the invariance, i.e., guarantee that

h(x(t)) ≥0,∀t≥t0

The process is summarized in Algorithm 1.

Complexity of Enforcing Invariance. The computational

complexity of the QP (16) is

O(q3)

, where

q=nP

K−1n

When there is a set

of output speciﬁcations, we just add

the corresponding constraint (10) for each speciﬁcation to

(16), and the number of constraints will not signiﬁcantly

increase the complexity. It is also possible to get the closed-

form solution of the QP (Ames et al.,2017) when there are

only a few output speciﬁcations.

4.2. Algorithms for Nonlinear layers

Stability of Auxiliary Systems. In this case, we need to

make sure that the parameter

θP

of the neural ODE is stabi-

lized as it is dynamically controlled by (11). To enforce this,

we use control Lyapunov functions (CLFs) (Ames et al.,

2012). Speciﬁcally, for each

θP

kj, j ∈ {1...,dP

, where

θP

is a component of

θP

, we deﬁne a CLF

V(θP

kj) =

(θP

kj−θP†

kj)2

, where

θP†

is the value of

θP

during or after

training. Then, any uP

kthat satisﬁes:

Φ(uP

k|θP

kj)≤0, j ∈ {1, . . . , dP

k},(17)

where

Φ(uP

k|θP

kj) = dV (θP

kj)

dθP

(AP

kjθP

k+BP

kjuP

k)+ϵjV(θP

kj)

kj∈R1×dP

k, BP

kj∈R1×dP

are the

’th rows of

k, BP

in (11), respectively and

ϵj>0

, will render the auxiliary

systems (11) stable. The proof is in Appendix A.3.

Enforcing invariance. Enforcing the invariance of a neu-

ral ODE is equivalent to the satisfaction of the condition

in Thm. 3.4. By proof in Appendix A.2, we can always

ﬁnd class

functions

α1, α2

such that there exists

that

makes (13) satisﬁed if

h(x(t0)) >0

. Again, we provide a

minimum-deviation quadratic program (QP) approach:

(uP∗

k,δ∗

1:dP

k) = arg min

k,δ1:dP

||uP

k||2+

j=1

wjδ2

s.t. (13) and Φ(uP

k|θP

kj)≤δj, j ∈ {1, . . . , dP

k},

(18)

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OntheForwardInvarianceofNeuralODEsWeiXiao1Tsun-HsuanWang1RaminHasani1MathiasLechner1YutongBan1ChuangGan2DanielaRus1AbstractWeproposeanewmethodtoensureneuralordi-narydifferentialequations(ODEs)satisfyoutputspecificationsbyusinginvariancesetpropaga-tion.Ourapproachusesaclassofcontrolbarrierfunctionsto...

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On the Forward Invariance of Neural ODEs

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