Physics-informed learning under mixing: How physical knowledge speeds up learning
Authors: Anna Scampicchio
Leonardo Felipe Toso
Rahel Rickenbach
James Anderson
Melanie Zeilinger
Venue: ICLR 2026 Poster (2026)
Keywords: learning with dependent data
physics-informed machine learning
convergence rates
complexity-dependent bounds
Primary area: learning theory
TL;DR: We prove that adding correct prior domain knowledge to nonparametric learning with dependent data speeds up learning

Abstract: A major challenge in physics-informed machine learning is to understand how the incorporation of prior domain knowledge affects learning rates when data are dependent. Focusing on empirical risk minimization with physics-informed regularization, we derive complexity-dependent bounds on the excess risk in probability and in expectation. We prove that, when the physical prior information is aligned, the learning rate improves from the (slow) Sobolev minimax rate to the (fast) optimal i.i.d. one without sample-size deflation due to data dependence.

Published as a conference paper at ICLR 2026
PHYSICS-INFORMED LEARNING UNDER MIXING: HOW PHYSICAL KNOWLEDGE SPEEDS UP LEARNING
Anna Scampicchio*
Leonardo F. Toso
Department of Electrical Engineering Chalmers University of Technology, Sweden anna. scampi och oe cha ra... se
Department of Electrical Engineering
Columbia University, USA
leonardo tosce co umbi a edu
Rahel Rickenbach
James Anderson
Institute for Dynamic Systems and Control ETH Zürich, Switzerland
Department of Electrical Engineering
Columbia University, USA james anderson co Lumbia edu
Irahel@ethz.c
Melanie N. Zeilinger
Institute for Dynamic Systems and Control ETH Zürich, Switzerland azeilinger@ethz.co
ABSTRACT
A major challenge thyyscc-inffrmme machine learning is to understand how the incorporation of prior domain knowledge affects learning rates when training data is not independent and identically distributed. Focusing on empirical risk mini- regularization,
mization with bounds on the excess risk in probability and in expectation. We prove that, when the physical prior information is aligned, the learning rate improves from the (slow) Sobolev minimax rate to the (fast) optimal i.i.d. one without sample-size deflation due to data dependence.
derive
INTRODUCTION
Physics-informed machine learning encompasses wide taxonomy of approaches that combine physical knowledge and learning algorithms to address two main tasks: (i) leveraging data-driven methods to enhance numerical solvability and accuracy of physical models given, for instance, by systems of partial differential equations; (ii) improve learning algorithm performance by including physical information in the loss function or as an additional constraint (Karniadakis et al., 2021; Meng al., 2025). Focusing the second class of methods, surveyed in Rai & Sahu (2020); von Rueden et al. (2023b), the resulting approaches turn out to be practically effective in terms of data efficiency, generalization capability and interpretability, which play an important role espe- cially in view of downstream tasks such as safe learning- based control (Nghiem et al., 2023; Drgona al., 2025). However, despite the empirical success rithms (Raissi et al., 2019; Rai & Sahu, 2020; Cuomo et al., 2022; Hao et al., 2023), theoretically quantifying the beneficial impact of incorporating physical information into learning algorithms is technically challenging and still an active area of research (see von Rueden et al. (2023a) and refer- ences therein).
mhysics-informen
machine learning algo-
We tackle this problem by considering statistical learning set-up and focusing regularized em- pirical risk minimization problems of the following form:
data-fit squared loss(
thysics-informet regularizer(F)
arg min ball Soboley space
>xr
(1.I)
"Corresponding author. A. Scampicchio and I.F. Taso share first-autorrship of this paper.

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where data entering the fit term dependent, derived from observations of a ground truth nonlinear dynamical system X1+1 = (X,) W,. with W, being a sub-Gausee Gaussian noise martingale difference sequence. The regularizer in (1.1) encodes the information that the true function to be estimated, f+, approximately satisfies a known partial differential equation induced by a linear operator 2 - i.e.,
we have that the regularizer takes the form || [ 22 and we say that knowledge alignment occurs if it holds that [ =0.
The main results this paper are complexiry-depencee [T+)) dr (Lecué & Mendelson, 2017) - for the excess risk || 107---1122 Informalyy.. our results (both in high probability and expectation) take the form given below:
bounds - i.e., bounds that depend on
Theorem (Informal). For suitable choice of the regularization parameter Xr, for suffi- ciently large number of samples T. and letting d < be the Sobolev minimax rate (Ibragimov & Has' minskii, 1981; Nussbaum, 2006), holds that
(Excess risk) |1f - f S Cslow (f.)
some power
noise level
+ Crast
T
T
The eπror bound established above reveals for the first time that, under knowledge alignment, the regularized estimate f converges to the true, unknown function f. at the i..dd rate of (1/TT)) in other words, behaves like classic optimal rates for i.i.d. learning even with dependent data after a suitable burn-in time.
The remainder of the paper unfolds as follows: Section 2 provides the set-up of the learning prob- lem, introducing the weighted, vector-valued function spaces that will be used throughout the paper. Next, the learning problem is stated in Section 3, and in Section 4 we provide the general state- ment for the excess risk bounds, both in probability and in expectation. Our analysis culminates in Section 5, where we prove how knowledge alignment leads to optimal i.i.d. rates even data are dependent. We discuss results in juxtaposition with related works in Section 6, and present some concluding remarks in Section 7.
2
PROBLEM SET-UP
This section collects preliminary concepts, defining the probability set-up of the data-ge setta-genrration mechanism (Section 2.1) 1) and the involved weighted, vector-valued function spaces (Section 2.2).
2.1 INPUT DOMAIN AND TRAJECTORY DISTRIBUTION
Let Ω [- Lixx C Ra be the input domain whose boundary is locally Lipschitz (Adams & Fournier, 2003, Definition 4.9). For horizon length T, the input trajectories denoted by X (Xy, X1, ..... Xx-1) belong to the metric space (NT { T- ,Px), where D = x
the Cartesian product of the single-component input domains Q; {x by a sequence of increasing a-algebras X++1 C X with respect to which X is adapted (Rogers & Williams, 2000, Chapter II.45); and Px the joint probability distribution of the input trajectory. As detailed in Section A.1, there exists a probability distribution associated with every component of we denote by f4, for each 0. ... T and we mostly work with a known initial distribution Ni for X, (typically, Dirac measure centered at the observed initial state Xo). Overall, we make use of the following:
-T- is the filtration given
Assumption 1. Let ya be the Lebesgue measure defined Ω C Rddx_ For all 0, each measure Jt:: Xi - T2zo is assumed to admit a density with respect to µĄ- We denote such density by (+))), and we assume that there exist g R < such that, for all f 0, . 1. < Pe R.
Note that Assumption accounts for many cases practical relevance, such as the uniform, trun- cated Gaussian, and beta distributions (Krishnamoorthy, 2016).
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2.2 SPACES OF FUNCTIONS
Space of square-integrable functions L2, We will focus on the Hilbert space L Px; Rdy) of vector-valued, square-integrable functions that consist of multiple evaluations of function f: Ω + Rdy along the input trajectory X. Such a space allows us to consider the trajectory X and is endowed with the inner product defined as follows: given f. g: Ω Rd,, we have
(f, 222 Rx (x)
Bpx (f((x)), e(xi))22
T-1
Σ
J (f(X)), g(Xi))2 dPx 1=0 O
T
T
T-1 (s(()), s(Xi)) veldXe),
Σ 1= 8
T
(2.1)
where ( 1)2 is the standard inner product defined in the Euclidean space Rdy, and H4 is the probability measure of the (-th component of X introduced in Section 2.1. The inner product (2.1) induces the trajectory norm (x)7 such that || (x
Px: == (J, 1) se P Moreover, it follows by construction that can write
|| [ 2 PxiR + E'= Epx [X))( Note in addition that, thanks to the separability of Ry the vector-valued space " (S"",Px+ Bd = {f: Ω + Rd llfll (XT.BBR)) < oo} 2 Px; R) (Conway, 2007, Chapter L6): indeed, ==
Px:B
can be written as the direct sum following (2.1), we can write
+ dy
dy Σ Bpx [A(X,]]] = dy I Px Px:
Σ 3
General L spaces. In general, one define the space L2( Br for any p Zzo endowed with the norm || (CT = + Et Ex (x))())) Of particular inter- est will be the Banach space of bounded functions _000 ( Rd))) equipped with the norm |lf g (or + supres fx)(2))
Sobolev space H.. Another fundamental function space derived from L2 Px;Rdr) is the multi-output, weighted Sobolev spuce of order Zzo. which is defined as follows:
H0 Px: = {f E (P::Rdy
|I (0T
00)
where the norm is induced by the inner product
(f, g) se (0.Px.:)))
=
Σ alS$
(D D 200TT Px:)
with D being the differential given by the multi-indes index a - (ai, adx) ) of non- non-negative inte- gers with order |a| = Ede 0;, i.e., D - 8 f/aaa/ adx Regarding the order of the Sobolev spaces we will consider, we will rely on the following:
Assumption The order of H. (12 Px; Bdy ) is non-negative integer that satisfies 2 2dxx
Finally, note that also the space H Ix; Rdy) admits the representation the direct sum t H (I2",, Px; R) thanks to the separability of Rdv This allows us to extend key results of show
scalar Sobolev spaces our vector-valued ones, detailed in Section B. In particular, that the Sobolev Imbedding Theorem (Adams & Fournier, 2003, Theorem 4.12) holds in our set-up, which will provide the necessary structure for the hypothesis space involved in the learning problem.
3 PROBLEM STATEMENT
Measurement model. Our data consists of trajectories of length T, denoted. D [Xe, Y generated according the measurement model
Y, - X fx(X,) + W.,
(3.1)
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where the noise sequence satisfies the following:
Assumption 3. The additive noise (Wi is martingale difference sequence with respect to
the filtration {&t hezz0 thus, Ew, [W][& for all = 0, ... ,T 1. Moreover, each W; is
also assumed to be oin-coniiionnalyy GabbSans given 11 i.., it holds that, for every 5 E R and every u the unit sphere in (Rdv, 11-112),
E [ 2
E exp (E (W,, 4)2 X & exp
(3.2)
The learning problem. In general, the learning problem can be stated as that of minimizing the excess risk |Iʃ f-ll R' searching for the estimate f within chosen hypothesis space F (which we specify later). However, since the underlying probability measures are unknown, the amount of data in D is finite and the hypothesis space & might be large, the estimate f is typically computed through (regularized) empirical risk minimization:
T-1
= arg min JE
Σ IIV s(x))1 (T)(()
(3.3)
T
Physics-informed regularization. The regularizer (.):) F - R20 encodes available prior physical information about the "true" function f. - in other words, (f) is designed to penal- ize the physical inconsistency between and the prior on f.. Such physical information is con- veyed by the fact that f. is assumed to approximately satisfy a known partial differential equation given by the linear operator 2: H.' (2, pxx; Rdy) + (S, px; Rdy). Such an operator is defined component-wiis as
12(f),
Σ Di,a fi for all i 1, dy. JolSs
(3.4)
where each pi + IR is a bounded function - therefore, if we denote by p the collection of all Pi then we have that |/p is finite. To describe the regularity of the differential operator in (3.4), we make the following assumption:
Assumption 4. The differential operator D() is elliptic that is, for all i = 1.... dy and any & E R2x {0). it holds that Σ =l== Pi,oS1 ... E. + 0.
Elliptic partial differential equations abound in practical applications, as they can be seen as gener- alizations of the Laplace and Poisson operators (Evans, 2010, Chapter 6). The differential operator z enters the definition of the regularizer in (3.3), where we have
V (f) = 19 (n Px BY)
(3.5)
which is a 2- 2-proper regularizer (Lecué & Mendelson, 2017, Assumption 1.1) - see Section E for the definition and further insights.
Hypothesis space. Let us now focus on the hypothesis space &. We consider it as the ball of radius P the Sobolev space, i.e.,
I = x² (x: IIfl (2T S Sp1).
(3.6)
Alternatively, as pointed out in (Cucker & Zhou, 2007, Theorem 8.21), one could write the cost in (3.3) as & - (Y, f(X.))² + would be performed for f ( PP Rdd), thanks to the equivalence yielding Ps ()())). In this paper, we make the following standing assumption:
Px + (J), and the minimization
Assumption 5. The hypothesis space f contains the unknown function to be estimated, f..
In case such an assumption is violated, the excess risk bound would feature an additional term quan- tifying the emrrr introduced by the mis-specification of the hypothesis space F. Such bias emror
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is entirely data- lata-independent and deterministic - therefore, it is decoupled from the analysis per- formed in this paper. In general, tools to bound the bias error rely on interpolation theory (Lunardi, 2018), and some results can be found, e.g., in Cucker & Smale (2002); Cucker & Zhou (2007).
Additionally, we will also consider the effective hypothesis space induced by the regularizer, namely
g = (S & V [a) p
(3.7)
For visualization of these hypothesis spaces, please refer Figure 1. Finally, we will sometimes
simplify notation by considering the shifted hypothesis space H. = H - f. = U - S- S H), with H. being for instance... I or F..
Modelling sample dependence in trajectories. Finally, we assume regularity the trajectory X given by the following one-sided exponential inequality (Samson, 2000):
Assumption 6. The trajectory X governed by the law Px in the hypothesis class & is S-persistent for some S E [1, x.). Specifically, for every & 2 0 and every I E $ we have that
]
T-1
25 Σ E t=
E exp
E t=0
exp
- Σ t=
B [1S(
Typically, S' is expressed in terms of the dependence matrix of X (see Section A.2 for its definition), and such a parameter attains higher values the more dependent X₃ is on its past. In general, S might depend on T; however, in this paper will focus the case in which S is constant: pointed out in (Samson, 2000, Section 2), this a rather weak condition satisfied by a large class of Markov chains and of ş--mxxng processes
Section A.2 for more details.
Contribution. Our results demonstrate that the physics-informed regularization in the empirical risk minimization problem (3.3) can speed-up the learning even in presence of dependent data. In particular, derive complexit-ddepender bounds for the excess risk || -of--all ( " both in probability and expectation, for learning under mixing, and prove that the rate of the excess risk matches the one from i.i.d learning in presence of knowledge alignment. Therefore, our results theoretically quantify the beneficial impact of physical knowledge learning algorithms, even in the challenging scenario of learning with dependent data.
BB()
F
Fo
Pf
Figure 1: Visualization of the involved hypothesis spaces. Note that the set 0B(r) = (/ E P. || :y = r²) introduced in Section 4.1 is represented as a square to highlight the fact that norm therein involved is different to the one defining & (3.6). Simi- larly, we represented F (3.7) as convex set that is not necessarily a ball in the Sobolev norm.
ERROR BOUNDS
We now present the bounds for the excess risk, both in probability and in expectation. We start in Section 4.1 by conveying the underlying ideas that lead to those results, and then provide the bound in probability (Section 4.2) and in expectation (Section 4.3). These results will be further analyzed in Section 5 to obtain our main claims the convergence rate of learning with physics- informed regularization. Before proceeding, we emphasize that the excess risk is a random quantity depending on the distribution the input sequence X and of the noises { [W.)-=+ therefore, often will simply write IP and E instead of Frx.w and Er w to streamline notation.
17-1;
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4.1 THE IDEA
The main idea consists of identifying an event according to which, with high probability and for some parameter 0,
Il
Px S - - fx(x))1
(4.1)
This kind of one-sided concentration inequality was studied for the i.i.d. setting in Mendelson (2014), to which we defer for a thorough discussion. The proof that (4.1) holds with high probability in the i..dd case is given in Mendelson (2014) thanks to the small-bal. condition, which weak assumption from a statistical point of view: see the discussion after Assumption 1.2 in Lecué & Mendelson (2017), together with its interpretation in terms of identifiability. In our data-dependent setting, the small-ball condition will be imposed by (C, tepprrecccottt with a = 2 (see Sec- r²) for any fixed > Therefore, the probability level of the event in (4.1) will be controlled by the radius r. We present a visualization Br)), together with all the hypothesis spaces, in Figure 1. Crucially, inequality (4.1) allows us to shift the analysis of the excess risk that of its empirical version. The next step consists then in upper-bounding the latter (i.e., the right-hand side in (4.1)) by the martingale offset complexity of the effective hypothesis space, Mr [F*"). In particular, for every f E F. (i.e., f f f. for some f' E P)), one has that
rather
tion D.2), and we show that it holds in the set (B(r) = {f E & Ilf --llle (xx
T-1
Σ 11(())1 < sup E (W, f(X,)) 11 Mr [ t=
(4.2)
FE#! T
We defer to Theorem G.1 for a derivation of such an inequality. Along the lines of Liang et al. (2015), we would like to stress that the term [I in the right-hand side introduces self-normalizing effect that compensates the fluctuations of the term (W,, f(X,))2. This fact is key in making the
martingale offset complexity not depend mixing, discussed in Section 5. One can provide bounds in probability and in expectation for the martingale offset complexity (see Section G), and these will play a key role in the excess risk bounds that present in the remainder of the section and further discuss in Section
Before presenting the aforementioned bounds, let formally introduce the lower isometry event, which is the complement of (4.1), whose probability we bound in Section F:
Ar
sup Berr)
1 T
Σ t=
(1(())1
| 11 1 $0
We are now in place to present the bounds for the excess risk, both in probability and in expectation.
4.2 RESULT IN PROBABILITY
Theorem 4.1. Let Assumptions I to 3, and 6 hold. Consider a parameter 0 > 8, and let f be the solution of the estimation problem (3.3) with Xy > 0, and let the radius p defining the effective hypothesis class F be such that p 2 (f.). Then, on the event
A AT
40
MT |$00|
3p
we have that
f 10
PMr ( + 2A (f.) 2
(4.3)
Proof. (Sketch). The proof follows Lecué & Mendelson (2017); Ziemann & Tu (2022) and it con- sists in characterizing the scenarios that lead to the event AC, showing that the case for which f F \ FP cannot occur for \r sufficiently large. The detailed proof is given in Section H.I. O

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4.3 RESULT IN EXPECTATION
Theorem 4.2. Let Assumptions 1 to 5 and 6 hold. Having the set 8B(r) that is (C(r), 2)- hypercontractive, let F, be rr/vv- in the infinity norm of 0B(r), and let Noo (DB(r), vo be the ussociated covering number. Consider the regularized empirical risk minimization problem in (3.3) with regularization parameter satisfying XT Z Bw [Mr [$ where p (f.)
Then, letting B be the positive constant such that ( S B for all f F, we have
E -
[|
fo
1
Noo (B((),
v0 + Vo (f+) +
r
exp
OCCSS
8T
OI M [
Proof. (Sketch). The idea consists in decomposing the expected value according to the lower-isometry event Ar and its complement: informally, E [excess risk nAA + E excess risk 0 nA... The first term would then be bounded thanks to S- persistence, (C, 2) tehyppercontractiiity and B- B-boundedness,, which allow us to quantify the probabil- ity of the lower-isometry event A, (see Section F). The bound for the second term is derived along the lines of the proof of Theorem 4.1. The full details are presented in Section H.2.
would write E [excess risk]
Overall, our analysis deploys the concepts of S S-persistence and (C, a.y.-yypercootrrctiiity to adapt the small-ball argument of Mendelson (2014) to the data-dependent case. Thanks to this construc- tion, we can identify the lower-isometry event, which enables the derivation of our bounds depending on the martingale offset complexity, the ground-truth regularizer (J()) and the critical radius r. In next section, we will characterize the behavior of these terms to obtain the desired convergence rates for physics-informed learning.
CONVERGENCE RATES
We finally provide our main results terms of convergence rates for the excess risk || - f+1 1-11(222 (X0 Ry whose detailed proofs are deferred to Section Building upon Theo- rems 4.1 and 4.2, we also obtain sufficient conditions on the parameter Xr and the minimal sample size 7 for the bounds to hold. Throughout this section, we will denote by d = 25/2s-dx the Sobolev minimax rate, and d' 2dx/2s+dx.
5.1 BOUND IN PROBABILITY
Theorem 5.1. Let Assumptions to 6 hold, and let J be the solution of (3.3). Fix a probubility of + + (Cn++riv) Cov
failure (0, 1). and assume the regularization parameter XT satisfies
Ciotta (ff.) 1(f.)1-4
+ Ci1or log(1/8)
Ar Z 37
(S.))
where CG CiI, Cirs and Ciy are constants depending only 8, dx, and Vlo((10)). the number of samples T satisfies
6d
da-dx
44 x=-
S
T2
CM
log
C+
+
log log(1/0) (1/8)
8
for = A (f.) (f.) + oly /T and Ch. Cm: Cr being uniform constants depending on PJ: F. 0, s, dx and Q, then, with probability at least 66, the excess risk enjoys the following convergence rate:
max (f.))'' 14 ((fa)dd
|5 1 sape Fx
o log(1/8)
I Csiov
+ Caat
where Cslow is constant that depends 8, dx, ,, dr, ow, sarvVolr /og(110), and Crast is a constant that depends on 8, dx, dy.

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Proof. (Sketch). The result builds upon the bound in probability the excess risk of Theorem 4.1, and its crux consists in conveniently setting the values for the critical radius т, the radius p of the effective hypothesis class FP, and the regularization parameter Ar. This allows to rewrite the excess risk bound (4.3) in terms of the martingale offset complexity, which can in turn be bounded according (Ziemann, 2022, Theorem 4.2.2) (see Theorem G.2 for its detailed proof). Finally, the characterization of the burn-in follows from the probability of the lower-isometry event. The full proof is reported in Section I.I, where the values of all of the involved constants are given.
5.2 BOUND IN EXPECTATION
Theorem 5.2. Let Assumptions I to 6 hold, and let f be the solution of (3.3) with regularization parameter Dr satisfying
4(CI+ Cc)(oC
XT Z
(f-)
where Cj and CTI are constants depending only 8, dx and dy. IfT satisfies
T 8
CRS
1
1xx 20
ox
CM
24
log
4B2
L+ (t)
2x
4x-dx
)
+ log
TW
T
where B is such that [l
2e= S B for all f E & and CM. Cn. CL are constants depending on P.: F. O, s, dx and Qt then the excess risk enjoys the following convergence rate:
E a|16
$1
[
f.
S C
(f=)
+ Crast
ow
Bx
T
T
where Czlow and Czast are constants that depend on s, dx, dy and ow.
Proof. (Sketch). Similarly to Theorem 5.1, one starts from Theorem 4.2 to set the values for p and Xr, and then deploys the bound on the expected martingale offset complexity of (Ziemann, 2022, Theorem 3.2.1) (see Theorem G.3 for its detailed proof). Ultimately, the claim is obtained by suitably choosing the critical radius and accordingly characterizing the lower-isometry event probability, leading to the expression for the burn-in. The detailed proof can be found in Section I.2.
Notably, our analysis allows us to transfer the contribution of data dependence from the excess risk bound to the burn-in time condition. Moreover, our bounds feature a fast, -.i.d.-like term (O(T-1)) and slower Sobolev rate term (O(T-A) that becomes annihilated when V (ff)) z O: this proves that, under knowledge alignment, the learning rate speeds up to ((T-1) even if data are dependent.
6 RELATED WORK AND DISCUSSION
General statistical learning framework. Statistical learning theory (Vapnik, 1998; Cucker & Smale. 2002) offers powerful tools to analyze the theoretical performance of nonparametric learn- ing algorithms. Within such framework, two main streams to derive learning rates have been developed, as identified by Fischer & Steinwart (2020). The first relies the spectral analysis of integral operators in reproducing kernel Hilbert spaces (Smale & Zhou, 2007; Caponnetto & De Vito, 2007; Steinwart al., 2009), while the second builds on empirical process techniques and the small-ball method (Mendelson, 2014; 2018; Lecué & Mendelson, 2017). Our work belongs to the latter stream, adapting the small-ball method to the dependent-dan data case along the lines of the localization analysis of Ziemann & Tu (2022).
Learning rates for dependent data. A common approach to handle dependence is through block- ing techniques (Yu, 1994; Sancetta, 2021), where the trajectory is divided into blocks of length k so that consecutive blocks can be treated as independent. However, this deflates the effective sam- ple size, leading to suboptimal rates. Similar rates appear also in Steinwart & Christmann (2009); Zou et al. (2009); Agarwal & Duchi (2012); Kuznetsov Mohri (2017). and Nagaraj et al. (2020) shows that such deflation in worst-case agnostic model set-up is unavoidable. To contrast this
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phenomenon, significant line of work has studied learning under dependent data without regular- ization. In the linear setting, Simchowitz et al. (2018) and Nagaraj et al. (2020) established sample complexity bounds for system identification and stochastic gradient descent. Moreover, Roy et al. (2021) extended the small-ball method to dependent processes, but without using one-sided con- centration, leading to slower rates. Similar slower-rate phenomena also appear in Ziemann et al. (2022). More recently. Ziemann& & Tu (2022) proposed an adaptation of the small-ball method and offistt complexity technique of Liang et al. (2015) to obtain optimal rates for nonlinear settings. Our work builds upon this line of thought, extending the analysis to However, the results in this paper are not mere adaptation: the physics-informer regularizer intro- duces additional challenges, such as characterizing the entropy numbers of the effective hypothesis class (e.g., under ellipticity, non-trivial nullspaces of the operator, and boundary conditions), deter- mining trajectory hypercontractivity and working with weighted, vector-valued Sobolev spaces.
regularization.
Theoretical analysis of physics-informed machine learning. Our work belongs to the branch of physics-informed machine learning that aims enhancing learning algorithms with available physical knowledge- a class of models also known as hybrid modeling (Rai & Sahu, 2020; Cuomo al.. 2022; von Rueden et al., 2023b; Hao et al.. 2023). To the best of the authors' knowledge, results aimed at quantifying the beneficial impact of physical priors in learning algorithms are von Rueden al. (2023a) and Doumèche et al. (2024). The present paper very similar in spirit to the latter work in the way deeeeeddtdd rates are derived, but crucially deals with non- i..d, data and presents bounds for the excess risk not only just expectation, but also in probability. We further summarize related work in Table 1.
Table 1: Comparison of convergence rates for non non-parametric regression with and without regularization, The rate from Ziemann & Tu (2022) follows from its Corollary 4.1 with q dx/s under the metric entropy bound (1/e)². The rate from Lecué & Mendelson (2017) follows from its Lemma 2.1 assuming r (0) ~ T with Ay ~ T
($,E) No
Work
Hypothesis class
Data
Regularization
Assumption
Rate
Nusshaum (20066
f Soholev space
lid. non-iid. iid non-lid.
2, Gaansiian dx
Farahmand & Sxepesvári (2012)
General Sobolev space General
Exponestial mixing, dy -1
T bogg)
Lecué & Mendelson (2017)
Proper regalarizer
sub Gaussian, dy
WIJ.TT T
Ziemann (2022)
General (noe too large)
Gaussian
Douméche et al. (2024)
Periadic Sobolev space
iid
informed
sub Gamma, dy
Vif.YT T-1
Our work
_ Soboley space
non-ii.
infseceddformeed
% sub Gaussian 2dx
+
Quantifying the impact of knowledge alignment. We now showcase the impact of knowledge alignment (f.)) = 0 in contrast with the rates of empirical risk minimization without regulariza- tion, i.e., considering f' as the solution of (3.3) when AT O. As shown in detail in Section the excess risk for f' behaves, both in probability and in expectation, in the following way (informally):
(Excess risk) I f-l (xIRM) S O'siON Tdd
+C
Ow T
We can notice how, for the result without regularization, the term decaying according to the Sobolev rate is not modulated by any design parameter (as happened with V (f.)) ) in Theorems 5.1 and 5.2), and is thus the dominant term dictating the slow Sobolev convergence rate of the excess risk.
On the behavior of Xr. It is worth emphasizing that, in both the expectation and probability analyses, the condition on the regularization parameter depends on 1 1/ for some / > 1. This condition reflects the well-known regularizationcompleeig trade-off:: as the hypothesis class is restricted (i.e., as p becomes small), one must increase AT to compensate for the reduced richness of the class and the potentially higher sensitivity noise variance, as discussed in (Lecué & Mendelson, 2017, Section 2) and also displayed in (Doumèche et al., 2024, Theorem 5.3). Even if such a phenomenon prevents us from considering the case (f..) O, our bounds still capture the (practical) annihilation of the Sobolev rate term in presence of knowledge alignment. Finally, as pointed out in Doumèche et al. (2024), even if \T depends on the unknown Vi (f+),, it can still be estimated in practice via, e.g., cross-validation (Wahba, 1990).
On the burn-in condition and the Sobulev order s. In Theorems 5.1 and 5.2, the burn-in time scales (1/r) odx/22 dx and in turn scales as T 1/2 Therefore, to ensure well-posedness of the

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burn-in time condition, we have to impose that 3dx/2s-dx I 1, which yields Assumption 2. Thus, our results come at the price stronger requirement on s with respect to the standard $ 2 dx/2 needed, e.g., for the Sobolev imbedding theorem (Section B).
Numerical illustration. We complement our theoretical analysis with an example showcasing the benefit of prior domain knowledge in learning a nonlinear dynamical system. In this experiment, whose full details can be found in Section K, we consider the dynamics of unicycle robot described by the differential equations
{
i (t) = (tt) cos (t). (t( vtt) sin ((t,,
0(() - (I),
where (21,22) E I2 is the position of the robot the plane, 2 [0, is the orientation angle, and (1,w) are the translational and angular velocities, respectively. The physical information want to incorporate is that the velocity has lateral component, enforcing the non-slip behavior of the unicycle kinematics. Such a constraint is embedded in the learning problem (3.3) a (discretized) L2 -ePrruulaaizztion term, and we perform estimation by deploying multilayer perceptron with two hidden layers featuring 64 nodes and ReLU activation functions.
The experiment, whose results are dis- played in Figure 2, compares the empirical rates obtained with and without physics- informed regularization. We can notice that both estimators eventually return an accurate model for the ground-truth dy- namics. However, without physics knowl- edge the rate of decay of the estimation error is relatively slow, with an empirical slope of approximately O(T 0.646). contrast, incorporating infsiee-informed regularization yields a markedly faster de- cay, with empirical slope of approxi- mately (TT-..993) as the model is ex- plicitly constrained by the domain knowl- edge that unicycle dynamics do not admit lateral velocity. This experiment demon- strates how embedding physics-based op- erators into the training objective leads to provable improvements in sample effi- ciency, consistent with the theory in Sec- tion 5 especially the result in expectation presented in Theorem 5.2.
knowledge (T 0.648)
with knowledge (T 0.993)
100
Estimation Error
10-1
10-2
10*
10a Number of samples
10a
10
105
Figure 2: Log-log plot of the empirical excess risk (estima- tion error) with respect the number of samples for the unicycle dynamics. Each curve obtained by averaging over 20 independent random realizations of the training data, with solid lines indicating the mean estimation error and shaded regions denoting 95% confidence intervals. The gray-shaded area displays the estimated burn-in time, after which the pre- dicted learning rates become observable.
7 CONCLUSIONS
This work focused sector-valueds function estimation from dependent data, and studied the excess risk of the estimate / obtained through regularized empirical risk minimization, where regularization is induced by physical knowledge (namely, that the unknown function approximately satisfies a partial differential equation). The analysis set in the general framework of statistical learning, and applies to a wide range of approaches such as kernel-based methods and (deep) neural networks with physics-inte -nffomeed loss functions. The main message of this work is that knowledge alignment (i.e., the regularizer is approximately zero when evaluated at the ground-truth function f+) allows to speed up the learning rate from the slow, Sobolev rate OT with the fast, optimal i.i.d. one O). Taken together, our results provide the first convergence rates for informed learning under dependent data that avoid the sample-so size deflation inherent to blocking techniques, and reveal transition from Sobolev minimax rates to fast i.i. rates through knowledge alignment. This bridges classical statistical learning theory, physics-infol informed regularization, and learning with dependent data.
2*/2s+dx < 1, to
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ACKNOWLEDGMENTS
Anna Scampicchio acknowledges that this work was partially supported by the Wallenberg AI, Au- tonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foun- dation. Leonardo F. Toso thanks Ingvar Ziemann for an instructive discussion during the early stage of this work. Leonardo F. Toso is funded by the Center for and Responsible Financial Innovation (CAIRFI) Fellowship and by the Columbia Presidential Fellowship. James Anderson is partially funded by NSF grants ECCS 2144634 and 2231350.
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Reviews:
[{"review_id": "eMfuLA0HVo", "reviewer": "Reviewer_tMrX", "summary": "This paper presents a theoretical analysis of physics-informed learning in the presence of dependent data. The authors analyze empirical risk minimization with a physics-informed regularizer that encodes known physical priors in the form of elliptic PDE constraints. Using the small-ball method and martingale offset complexity, they derive complexity-dependent excess risk bounds both in probability and expectation. The main result shows that when the physical prior regularizer is aligned with the true dynamics, the convergence rate improves from the slow Sobolev minimax rate to the optimal i.i.d. rate, even with dependent samples. The theoretical framework is supported by a unicycle dynamics experiment illustrating the empirical benefit of incorporating physics-informed regularization.", "questions": "1. see in the weakness\n\n2. The phsyics-informed regularizer is limited to linear elliptic PDEs. However, the data is dependent, which is always occured in dynamical systems. Is this two conflicting? The numerical example is an ODE, not elliptic PDEs. How the convergence rate behaves for common PINNs problems such as Poisson equation or Darcy flow problem? How can the proposed theory be connected to the optimization landscape of neural networks trained with PINNs?\n\n3. Does the empirical rate in the numerical example persist when neural architectures differ from MLPs, or with stochastic training?\n\n4. Can the presentation be more friendly to the general ICLR audience without additional intuition or graphical explanation of the key proof mechanisms? The current form is mathematically dense and more suitable to journals like JMLR, not general top conferences.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper rigorously extends the small-ball method and offset complexity analysis to dependent data settings with physics-informed regularization, which is technically novel.\n\n2. The paper establishes a clear theoretical connection between physical priors (encoded as elliptic PDE constraints) and improved learning rates, addressing a long-standing gap in understanding the benefits of physics-informed models.\n\n3. The proofs and appendices appear detailed, well-grounded in functional analysis, Sobolev space theory, and dependent process theory.\n\n4. The  experiment, though simple, can demonstrate the theoretical prediction that physical priors accelerate convergence.", "weaknesses": "1. The analysis assumes elliptic linear PDE operators; it is unclear whether the results generalize to non-elliptic, nonlinear, or mixed-type operators often seen in physics-informed neural networks (PINNs). The gap or difficulty is not well addressed. \n\n2. The optimization error's influnene is not discussed in the theoretical part and the numerical example. Note that the physics-informed regularizer will increase the stiffness of the Hessian matrix and increase difficulties for the optimization in PINNs, which is not aligned with the main result (convergence rate improves with physics-inforemed regularizer added)\n\n3. Too many assumptions may not hold for complex real-world systems, a more interpretable or verifiable condition would strengthen the practical relevance.\n\n4. Only one low-dimensional example is provided. Additional tests on nonlinear PDE systems or stochastic dynamical systems would reinforce the claims.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FZP46QPJNB", "reviewer": "Reviewer_3SEU", "summary": "The paper derives complexity-dependent bounds that formally characterize the impact of aligned prior domain knowledge on learning rates in physics-informed machine learning settings. This is approached by focusing on bounds of the excess risk in regularized ERM under dependent data generated from non-linear dynamical systems. Results under the combined assumptions of physics-informed regularization and non-IID data have yet to be shown in existing literature, and the derived bounds reveal new insights into the relationship between statistical learning theory and physics-informed ML.", "questions": "- Are there any promising avenues for achieving the stronger burn-in requirement while relaxing $s\\ge 2d_X$ toward the standard $s\\ge d_X/2$, or are there clear reasons to believe this is a necessary tradeoff? Can factors/assumptions be loosened in its place while maintaining the original bound, giving another route to the same result?\n- In the numerical experiment and Figure 2, does the reported slope fit account for burn-in (if even needed in this case)? It would be instructive to see how this shows up here, and/or when burn-in exceeds realistically attainable sample sizes in practice.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The paper is well-organized and has a presentation that's easy to follow from start to finish. I appreciated the clear groups in Section 3 and the visual distinction for all stated assumptions.\n- The authors address an important theoretical challenge in the physics-informed ML space, with a fundamental connection to all PIML architectures and practitioners looking to tackle scientific problems. Characterizing the impact of well-aligned domain knowledge takes a clear step toward helping researchers and practitioners alike when it comes to justifying additional time spent refining assumptions and calibrating inductive biases.\n- Relaxing the IID assumptions from prior work (e.g., Doumèche et al. 2024) appears to be an important adjustment that broadens the reach of the resulting bounds. In most PIML settings, one is working with heavily dependent sequential data, so this seems a prudent step toward establishing useful bounds that mirror the real world.", "weaknesses": "- Assumption 5 is quite strong, limiting the applicability of the results in many real-world physics-informed modeling scenarios. For instance, unless additional constraints are applied, popular methods like physics-informed neural networks would presumably violate this assumption, and in these cases it's unclear how one should think about the applicability of the results. I understand details on approximations were stated as out of scope, but it would be nice to include some discussion/analysis of the likely ways practitioners may violate assumptions in practice and what elements of the original bound still apply (if any). \n- Similar to the above point, Assumption 4 also excludes common practical scenarios that leverage non-linear PDE priors. Discussion on what remains of the bound in these cases would be helpful for position the paper's analysis in a broader context.\n- This is somewhat tangential to the core theoretical aims of the paper, but it would be instructive to include a more holistic case study. In particular, the impact of using misaligned/incomplete prior physics knowledge and how this empirically threads between the \"with knowledge\" and \"without knowledge\" scenarios. The sample sizes at which various levels of knowledge alignment outperform others, if only for short ranges of $T$, would also be helpful to characterize insofar as they relate to the paper's analysis of sample size.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ms1clVzy5v", "reviewer": "Reviewer_6PJM", "summary": "This paper develops a rigorous statistical-learning theory for physics-informed learning under dependent (mixing) data. It studies regularized empirical risk minimization where the regularizer encodes known physical laws through a linear differential operator, and the data arise from a stochastic dynamical system $X_{t+1}=f_{\\star}(X_t)+W_t$. Using tools from Sobolev-space analysis, the small-ball method, and martingale offset complexity, the authors prove complexity-dependent excess-risk bounds showing that when the physical prior is aligned with the ground-truth dynamics (i.e., PDE residual of $f_{\\star}$ is nearly zero), the learning rate acclerates from the traditional Sobolev minimax rate $O(T^{-d})$ to the fast i.i.d optimal rate $O(1/T)$, even when samples are correlated. A simple unicycle-dynamics experiment empirically confirms the predicted speed-up, demonstrating that properly aligned physics-based regularization can provably improve sample efficiency in learning dynamical systems.", "questions": "In relation to the weaknesses stated above, please see my questions/comments below:\n\n1. The paper presents a two-term bound combining the $T^{-d}$ and $T^{-1}$ rates, but it is unclear how to interpret the intermediate regime. Could the authors clarify whether the transition between the two regimes is smooth or abrupt, and provide any threshold conditions under which the fast term becomes dominant? In addition, it'd be great if the authors could shed some light on the following: how robust this behavior is to imperfect knowledge alignment? For example, if $\\|\\|D\\left(f_{\\star}\\right)\\|\\|_{L^2}$ is small but nonzero, does the convergence rate degrade gradually (and remain faster than $T^{-d}$ ) or does it collapse sharply to the slower regime?\n\n3. Since $\\Psi(f)=\\|\\|D(f)\\|\\|_{L^2}^2$ may involve higher-order derivatives, do the authors foresee computational or numerical challenges when scaling beyond low-dimensional synthetic problems? Any guidance on discretization or numerical stability when implementing this regularizer in practice would be helpful.\n\n3. The analysis assumes that $D$ is a linear and elliptic operator, which can be restrictive. Could the authors comment on whether any part of the analysis may extend to mildly nonlinear or non-elliptic operators, or if linear ellipticity is fundamentally required for the proof techniques used?\n\n4. The paper imposes $s \\geq 2 d_X$ to ensure the burn-in term vanishes. Is this threshold believed to be intrinsic to the problem, or could the two-phase rate behavior persist under weaker smoothness assumptions (e.g., $s>d_X$ or $s>3 d_X / 2$ ) with possibly different constants?\n\n5. Could the authors provide any preliminary numerical results or intuition on how large $T$ must be for the asymptotic behavior to manifest in practice? Even a brief discussion of the finite-sample regime would help readers assess when the theoretical rates become observable.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper addresses an important topic at the intersection of physics-informed learning and statistical learning under dependence. The authors aim to provide a theoretical foundation for when incorporating physical structure can mitigate the challenges of temporally correlated data. In this context, the paper offers several notable positive aspects:\n\n1. The central idea of using elliptic differential operators to encode physical priors and recover i.i.d.-like learning rates (under suitable alignment) despite sample correlation is novel and insightful.\n\n2. The theoretical analysis is mathematically sound. The assumptions are clearly stated, and the main theorems follow logically from the lemmas and proof techniques provided. The paper carefully extends existing theory to accommodate a Sobolev-based physics regularizer in the presence of Markovian data, which is nontrivial. While I did not look into the proofs in detail and some proof components rely on established techniques, the overall combination is coherent and technically competent.\n\n3. The results contribute to a better theoretical understanding of physics-informed learning, particularly in settings where data dependence is unavoidable, such as system identification and scientific modeling. Showing that incorporating correct physical structure can mitigate the adverse effects of temporal correlation is a useful insight, although demonstrated under idealized assumptions.", "weaknesses": "While the paper makes a meaningful theoretical contribution and is clearly written, several limitations temper its overall impact and practical relevance. Most of these relate to the idealized nature of the assumptions and the gap between the theory and empirical applicability. The following points highlight areas where the work could be strengthened or where additional clarification or experimentation would improve the contribution:\n\n\n1. The fast-rate $O(1 / T)$ convergence is achieved only under an idealized knowledge alignment condition, namely when $\\|\\|D\\left(f_{\\star}\\right)\\|\\|_{L^2} \\approx 0$. In practice, the physical operator $D$ is rarely known with such precision, and even modest mismatch can revert the rate to the slower $O\\left(T^{-d}\\right)$ regime. The paper does not provide a quantitative robustness analysis that would clarify how sensitive the rates are to partial or imperfect alignment, which limits the applicability of the theoretical claims in realistic settings.\n\n2. The theoretical guarantees depend on $\\lambda_T$ choices tied to latent problem quantities (e.g., $\\Psi(f_{\\star}), \\sigma_W^2$ ). While the authors note that cross-validation could be used in principle, the paper offers limited practical guidance or empirical validation for tuning $\\lambda_T$ in realistic settings.\n\n3. The single toy experiment (unicycle) is supportive but narrow (one setting, small MLP, no real-world data/baselines), so the robustness of the phase transition across architectures/noise regimes remains unclear. Even in this controlled synthetic setting, implementing the physics-informed regularization term $\\Psi(f)= \\|\\|D(f)\\|\\|_{L^2}^2$ may require evaluating higher-order derivatives and Sobolev norms, which can be computationally demanding in higher dimensions. In addition, the paper does not discuss how discretization, numerical differentiation, or instability in PDE solvers or neural approximators would affect the performance or validity of the theoretical bounds, leaving a gap between the continuous theory and practical implementation.\n\n4. The framework relies on $D$ being a known, linear elliptic operator. Many modern scientific machine learning applications involve unknown or nonlinear physics, or operators that must be learned jointly with the model (e.g., operator learning, neural PDE surrogates, and PINNs). As the current analysis does not extend to such settings, it is unclear how the insights would generalize to applications where the governing equations are only partially known or inherently nonlinear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pNqICDZ6FG", "reviewer": "Reviewer_7BGW", "summary": "This manuscript presents a theoretical analysis of physics informed machine learning with data are dependent. The work is an attempt to address the open challenge why incorporating physics prior can benefit data-drive learning.", "questions": "1. Minor formatting issue: all equations should be numbered.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. This paper addresses the open challenge: the theoretical soundness of physics informed ML, beyond empirical evidence and intuitive understanding that the prior physics knowledge is conductive to learning. \n2. Theoretical analysis appears to be rigorous (but I didn't check all the proofs).", "weaknesses": "1. The organization and exposition of the manuscript can be improved. Without loss of theoretical rigor, it would be great if the intuitive explanations can be provided for deep learning practitioners the meaning of the theoretical results in practice. \n2. Related to the first question, intuitive understanding of the importance of conditions related to key variables such as $T$, as in Theorem 5.2 and $\\lambda_T$ as in theorem 5.1, would greatly benefit the reader.\n3. It's understandable this is a theoretical paper, but the only numerical experiment does add the weight to the paper. A well thought-off experiments to demonstrate the conditions related to $T$ and $\\lambda_T$ would also greatly benefit the readers, see the concern above.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)