Space-time error estimates for deep neural network approximations for differential equations

Arnulf Jentzen; Fabian Hornung; Philipp Grohs; Philipp Zimmermann

Space-time error estimates for deep neural network approximations for differential equations

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read ReLU networks approximate Euler trajectories in time and space.

desk verdict Solid, fully written-up first space-time error estimates for ReLU-network Euler schemes; the central proof holds up, with the usual caveat that it is about Eulerization, not PDEs. read the letter →

arxiv 1908.03833 v1 pith:VFEXWTVF submitted 2019-08-11 math.NA cs.LGcs.NA

Philipp Grohs , Fabian Hornung , Arnulf Jentzen , Philipp Zimmermann This is my paper · ORCID

classification math.NAcs.LGcs.NA MSC 65D1568T07

keywords deepneuralnetworksReLUEulerapproximationspace-timeerrorestimatesANNcalculusdifferentialequationsrectifiedlinearunitstheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that rectified deep neural networks can approximate the space-time evolution of a large class of differential equations: for any prescribed tolerance, dimension, and number of Euler steps, there is an explicit ReLU network whose output is uniformly close to the Euler approximation of the solution across the whole time interval and all of R^d, not just at a fixed time. The error is controlled simultaneously in space and time by a single parameter epsilon, with the network parameter count bounded by an explicit polynomial in the number of Euler steps and a logarithmic factor in 1/epsilon. This closes a gap left by earlier rigorous results, which gave spatial error estimates for neural-network PDE approximations but no temporal component. If the estimates hold, they supply a first rigorous consistency check for using deep networks as space-time solvers rather than as spatial function approximators.

What carries the argument

The argument is carried by an ANN calculus, a compositional algebra for building networks realizing sums, parallelizations, compositions, and Euler updates, together with explicit ReLU approximations of the square function x -> $x^{2}$ and of the product (t,x) -> tx. The product network is the space-time bridge: it converts each Euler step into a network operation, hat functions interpolate between grid times, and a discrete Gronwall estimate keeps the trajectory growth polynomial in ||x||. The rectifier activation a(x)=max{x,0} is used throughout.

What would settle it

Check the claim for d=1, N=1, T=1, with drift R(Phi)(x)=max{x,0}. The Euler map is Y_{t,x}=x+t*max{x,0}, so the network must realize the piecewise-quadratic map (t,x) -> x+t*max{x,0} on [0,1] x R. Compute the minimal ReLU network parameters needed to approximate this map to error epsilon and compare with the asserted bound P(Psi) <= C(1+|ln(epsilon)|^2); if the required parameter count grows faster than log(1/epsilon) as epsilon tends to 0, the epsilon-dependence in Theorem 1.1 fails.

Watch

Extended reading notes

Core claim

The main result, Theorem 3.12 with the special case Theorem 1.1, asserts that for every epsilon in (0,1], every dimension d, and every number N of Euler steps, there exist ReLU networks $Psi^{{epsilon,d,N}}$ such that for all t in [0,T] and x in R^d, ||$Y^{{d,N}}$_{t,x} - R($Psi^{{epsilon,d,N}}$)(t,x)|| <= C $d^{{1/2}}$ $N^{{3/2}}$ epsilon (1+||x||^3), while the network size satisfies P($Psi^{{epsilon,d,N}}$) <= C $d^{{16+8d}}$ $N^{6}$ (1+|ln(epsilon)|^2). Here $Y^{{d,N}}$ is the Euler approximation of an ordinary differential equation whose drift is itself given by a network with linear growth. The approximation is space-time in the strong sense: the same network realizes the solution map (t,x) -> Y_{t,x} continuously on [0,T] x R^d, and the error is uniform in t and x.

Load-bearing premise

The drift network must grow at most linearly, with ||R(Phi)(x)|| <= C(1+||x||), because that bound is the input to the Gronwall step that keeps the Euler trajectories finite and makes the a priori bounds enter the error estimate; if it is violated, the claimed uniform bounds can blow up.

Editorial extensions

If this is right

For every tolerance epsilon, dimension d, and number of Euler steps N, a ReLU network of explicitly bounded size realizes the Euler map uniformly on [0,T] x R^d.
The time direction is controlled by the same accuracy parameter epsilon as the spatial direction, closing the gap left by purely spatial estimates.
The construction is explicit enough to be read as an algorithm: compose networks for the square, the product, hat functions, and the Euler update.
The network size grows polynomially in N and logarithmically in 1/epsilon, but exponentially in the dimension d.
The a priori bounds used in the proof are uniform in the perturbation sequence y, so the same network family covers perturbed Euler schemes in one stroke.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper stops at Euler approximations of ODEs; the announced next step is to feed these space-time networks into PDE approximation schemes. A testable extension is to insert the constructed network into a splitting or backward-solver pipeline and check whether the observed error obeys the same epsilon-dependence.
The size bound d^{16+8d} N^6 (1+|ln(epsilon)|^2) grows exponentially in d, so the result is a consistency and expressivity statement for fixed dimension, not a curse-of-dimensionality breakthrough; a follow-up would need to reduce this to a polynomial-in-d bound.
Because the construction is explicit and deterministic, it provides a training-free baseline for neural emulation of differential-equation solvers: the weights can be hard-coded and evaluation cost compared against classical Euler integration.
The requirement that the drift network grow at most linearly is inherited from the Gronwall step; weakening it would likely need a different a priori control and would change the polynomial-in-N factors in the error bound.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: the main result is real and it is new. Prior work on ReLU approximation of PDE-related quantities only had spatial estimates; this paper gives the first space-time estimates, applied to Euler schemes for ODEs, with explicit rates and network-size bounds. I checked the main chain and the proof is sound. The ANN calculus (compositions, parallelizations, sums) is developed carefully, the product approximation is built correctly from the square approximant, the hat-function interpolation gives a clean a posteriori bound, and the discrete Gronwall argument converts it to an a priori bound without any hidden assumption. The paper also genuinely improves on its predecessor [25] in the parameter estimates of Lemma 2.27(iv) and Proposition 2.28(iv).

The most likely worry is the linear-growth assumption on the drift network. The reader flagged it as the weakest premise, and the stress-test note says it is automatic for fixed-depth ReLU networks. The stress-test note is right. A fixed ReLU network is piecewise linear and therefore globally Lipschitz, so the bound ||R(Phi)(x)|| <= C(1+||x||) costs nothing. The constant C only enters through exp(CT), which is exactly what a Gronwall argument should produce. No fragile cancellation is needed.

Soft spots, in proportion: the estimates are for the Euler-approximated trajectory, not for the solution of the underlying ODE or PDE. The authors state this plainly in the introduction and even say the PDE transfer is future work. That is a real limitation for somebody hoping for a ready-made PDE result, but it is not a flaw in the theorem as stated. The parameter bound in Theorem 1.1 has d^{16+8d}, which is unusably large in high dimension; this is a qualitative contribution, not a practical recipe. The paper is also extremely notationally heavy, with every elementary composition lemma proved in full. That is a stylistic choice, costly for the reader but not wrong.

The citation pattern is heavily self-referential, but the relied-on results from [25] are independently proved and are not restatements of the target theorem. No circularity. No code or data, which is standard for this kind of proof-driven paper.

Who gets value: researchers in DNN approximation theory and numerical analysis who want a template for space-time estimates, plus anyone working on the Jentzen group's program. Practitioners will not find this useful.

My verdict: this deserves a serious referee and likely acceptance after minor revisions. I would send it out without hesitation. The result is a correct, reusable step forward, and the limitations are explicit rather than hidden.

Referee Report

0 major / 4 minor

Summary. The paper claims to provide the first space-time error estimates for deep ReLU network approximations of Euler approximations of perturbed differential equations. The main result, Theorem 3.12, asserts that for an Euler scheme driven by a ReLU drift network with linear growth, there exist ReLU networks whose space-time map approximates the Euler flow with explicit error bounds of the form eps times a polynomial in d, N, and the a priori growth bound, together with explicit parameter-count bounds polynomial in N and d and quadratic in |log eps|. The proof develops an ANN calculus for compositions, parallelizations, and sums of ReLU networks; constructs product-approximation networks from a square-function approximant; and combines a posteriori error estimates via time hat-function interpolation with a discrete Gronwall a priori bound. Corollary 3.13 and Theorem 1.1 give simplified closed-form bounds, including the y=0 case stated in the introduction. The paper explicitly states that the estimates concern Euler approximations rather than the differential equation or PDE solution itself.

Significance. If correct, the result is a useful foundational contribution: it appears to be the first space-time error estimate, rather than a purely spatial one, for DNN approximation of an Euler scheme, and it supplies reusable ingredients with fully written proofs and explicitly tracked constants. The central statement is a conditional existence result with no free parameters: given any ReLU drift network, the constructed network and the error and size bounds are explicit. The weakest-assumption concern raised in review, namely the linear growth bound on the drift network, is not fragile: every fixed-depth ReLU network is globally Lipschitz, so the bound holds automatically with C equal to the network's Lipschitz constant, and the constant only enters through exp(CT). The main limitation, that the result concerns Euler approximations and not the exact ODE/PDE solution, is stated honestly in the abstract and introduction and does not undermine the central claim.

minor comments (4)

[Proposition 3.5(v), Section 3.2.1] The displayed intermediate bound in Proposition 3.5(v) appears as "+ 1/(q-2) - 252", while the proof around Eq. (306) establishes the same expression with "+ 720/(q-2) - 252". The final bound is unaffected, but the statement should be corrected for consistency.
[Section 2 and Proposition 3.10] The paper says that every result in Section 2 is proved in detail, yet Proposition 3.10 invokes the existence of identity networks I_d with D(I_d)=(d,2d,d) and R(I_d)=id by citing [25, Lemma 5.4], and Lemma 2.14 treats such networks as an assumption. Since the construction is elementary, either include it or explicitly acknowledge this external result in the self-containedness statement.
[Theorem 1.1 and Corollary 3.13] The final constant is stated as "C in R", but the displayed inequalities are only meaningful when the constant is nonnegative; the statements should read "C in [0, infinity)".
[Title and abstract] The title refers to "differential equations", but the theorem concerns Euler approximations rather than solutions of the differential equation; although the abstract and introduction make this clear, the title could be more precise, e.g., by mentioning Euler schemes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is a fully self-contained conditional existence result.

full rationale

I walked the derivation chain from the ANN calculus in Section 2 through the square-function and product approximations in Section 3.1–3.2 to the a posteriori estimate (Proposition 3.10), the Gronwall-based a priori bound (Lemma 3.11), and the main Theorem 3.12. Each step is proved in the paper with explicit constants, and the final estimate is a genuine conditional existence statement: given any ReLU drift network Φ with linear growth, the paper constructs a ReLU network Ψ whose realization approximates the Euler scheme of Φ in space-time with explicit error and parameter bounds. No parameter is fitted to the target quantity Y, and no equation is used as both input and output. The square-function approximant is built from an explicit hat-function interpolation (Lemmas 3.1–3.2), the product approximant from the polarization identity and the square approximant, and the Euler representation from a hat-function interpolation of the trajectory; all of these are independent constructions rather than restatements of the desired estimate. The paper does cite prior work from the same research group, notably [25] for an elementary identity-network lemma and for inspiration in the Euler representation, but the relevant identity network is a trivial ReLU construction and the Euler-representation argument is re-proved in the paper (Proposition 2.32 and Corollary 2.31). Such citations are not load-bearing in the sense of importing the target theorem. The acknowledged limitation—that the estimates concern Euler approximations rather than the differential-equation solution itself—is stated in the abstract and introduction and does not create circularity. No self-definitional step, fitted-input-called-prediction step, or imported-uniqueness argument was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper proves its results from standard mathematical background and a few explicit hypotheses. It relies on the discrete Gronwall inequality and on prior constructions of ReLU identity networks and square function approximations, all cited with proofs in the literature. The drift growth bound and the size bound on the drift network are stated hypotheses, not derived results. No free parameters are fitted to data, and no new entities are postulated.

assumptions (5)

standard math Discrete Gronwall inequality (Hutzenthaler et al. [23, Lemma 2.1])
Used in Lemma 3.11 to obtain the a priori bound on the Euler trajectory; standard result cited from prior literature.
standard math Existence of a ReLU identity network I_d with D(I_d)=(d,2d,d) and R(I_d)(x)=x (Jentzen-Salimova-Welti [25, Lemma 5.4])
Used in Proposition 3.10 to extend networks to a common depth; proved in cited prior work, not in this paper.
standard math ReLU approximation of the square function on [0,1] with error ε and O(log ε^{-1}) parameters (Yarotsky [37], Elbrächter et al. [11])
Basis for the product approximation results in Propositions 3.3-3.7; the paper reformulates and proves the needed versions, but the original technique is due to prior work.
domain assumption Growth bound on the drift network: ||R(Φ)(x)|| ≤ C(1+||x||)
Stated as a hypothesis in Theorem 3.12 and used in the Gronwall estimate in Lemma 3.11; without it the trajectory bound g_n fails.
domain assumption Size bound on the drift network: P(Φ_d) ≤ C d^d in Corollary 3.13 / Theorem 1.1
Stated as a hypothesis; it enters the final parameter bound P(Ψ) ≤ C d^{16+8d} N^6(1+|ln ε|^2).

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Cite this review

Pith. "Pith review of Space-time error estimates for deep neural network approximations for differential equations." pith.science (2026). https://pith.science/paper/VFEXWTVF

@misc{pith2026190803833,
  author       = {Pith},
  title        = {Pith review of: Space-time error estimates for deep neural network approximations for differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFEXWTVF}},
  note         = {Machine review of arXiv:1908.03833}
}

abstract

Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of partial differential equations (PDEs) by means of stochastic learning problems involving DNNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for neural network approximations for PDEs but do not provide error estimates for the entire space-time error for the considered neural network approximations. It is the subject of the main result of this article to provide space-time error estimates for DNN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain artificial neural network (ANN) calculus and (ii) on ANN approximation results for products of the form $[0,T]\times \mathbb{R}^d\ni (t,x)\mapsto tx\in \mathbb{R}^d$ where $T\in (0,\infty)$, $d\in \mathbb{N}$, which we both develop within this article.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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