arxiv: 2604.05187 · v1 · submitted 2026-04-06 · 💻 cs.LG · cs.SY· eess.SY

Recognition: 2 theorem links

· Lean Theorem

FNO^{angle θ}: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems

Zhexian Li , Ketan Savla

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:16 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SY

keywords Fourier neural operatorcomplex frequencyEhrenpreis-Palamodov principlePDE optimal controlBurgers equationdistributed parameter systemsneural operatorslinear quadratic control

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The pith

Extending the inverse Fourier transform in FNO to complex frequencies captures integral representations of PDE states and controls.

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

The paper shows that states and optimal controls of linear constant-coefficient PDEs admit an integral representation in the complex domain via the Ehrenpreis-Palamodov fundamental principle, where the integrand shares the exponential kernel with the inverse Fourier transform. This observation leads to a direct modification of the FNO layer: the frequency variable in the inverse transform is extended from the real line into the complex plane. The resulting architecture is evaluated on the nonlinear Burgers' equation for simultaneous learning of state evolution and linear-quadratic optimal control, where it produces substantially lower training errors and more faithful non-periodic boundary predictions than the original FNO. Readers interested in distributed-parameter control would care because the method offers a structured way to embed analytic representations of linear PDE solutions into neural operators that must also handle nonlinearity.

Core claim

Any state and optimal control of linear PDEs with constant coefficients can be written as an integral over the complex domain whose integrand contains the same exponential term appearing in the inverse Fourier transform; therefore extending the frequency variable of that transform inside each FNO layer from the real to the complex domain directly encodes the fundamental-principle representation and yields improved accuracy even when the target system is nonlinear.

What carries the argument

FNO layer whose inverse Fourier transform is performed over a complex frequency variable, directly implementing the complex integral representation supplied by the Ehrenpreis-Palamodov principle.

If this is right

Order-of-magnitude reduction in training error for learning both state and linear-quadratic optimal control on Burgers' equation.
Improved accuracy in predicting non-periodic boundary values compared with the original FNO.
The same layer modification applies uniformly to linear constant-coefficient PDEs and to selected nonlinear systems.
The architecture supports joint learning of state trajectories and additive optimal controls without requiring repeated PDE solves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same complex-frequency modification may improve neural-operator accuracy on other nonlinear PDEs whose linearizations admit complex-integral representations.
Control policies learned with the extended operator could be more robust to changes in boundary conditions than those learned with real-only FNO layers.
The method supplies a concrete route for embedding known analytic structure of linear PDEs into operators that must handle nonlinearity, suggesting hybrid designs for higher-dimensional distributed systems.

Load-bearing premise

The complex-frequency extension derived for linear constant-coefficient PDEs transfers effectively to learning state and optimal control on nonlinear PDEs such as Burgers' equation.

What would settle it

Training both the extended and standard FNO on the same Burgers'-equation state-and-control task and finding that training errors do not drop by an order of magnitude or that non-periodic boundary predictions remain no more accurate would falsify the performance claim.

Figures

Figures reproduced from arXiv: 2604.05187 by Ketan Savla, Zhexian Li.

**Figure 1.** Figure 1: Comparison of relative mean squared errors (MSE) over all training data during the training phase of FNO and FNO∠θ for the Burgers’ equation (16). FNO∠θ achieves order of magnitude improvement over FNO. training. The training dataset consists of 50 state trajectories under different sampled boundary conditions. For both FNO and FNO∠θ , we use the Gaussian Error Linear Unit (GELU) activation function for σ … view at source ↗

**Figure 2.** Figure 2: Learning state operator: comparison of the absolute error between the neural operator output and the training data for a particular boundary condition. The error for FNO is concentrated on the boundaries, whereas FNO∠θ generates more accurate boundary values. Figure 1a compares the relative mean squared errors (MSE) during the training phase of FNO and FNO∠θ for learning state operator. The values of rel… view at source ↗

read the original abstract

We propose an extended Fourier neural operator (FNO) architecture for learning state and linear quadratic additive optimal control of systems governed by partial differential equations. Using the Ehrenpreis-Palamodov fundamental principle, we show that any state and optimal control of linear PDEs with constant coefficients can be represented as an integral in the complex domain. The integrand of this representation involves the same exponential term as in the inverse Fourier transform, where the latter is used to represent the convolution operator in FNO layer. Motivated by this observation, we modify the FNO layer by extending the frequency variable in the inverse Fourier transform from the real to complex domain to capture the integral representation from the fundamental principle. We illustrate the performance of FNO in learning state and optimal control for the nonlinear Burgers' equation, showing order of magnitude improvements in training errors and more accurate predictions of non-periodic boundary values over FNO.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Complex-frequency extension to FNO is motivated for linear constant-coefficient PDEs but applied as a heuristic to nonlinear Burgers' control, with reported large error drops that need parameter-controlled checks.

read the letter

The main takeaway is that this paper modifies the FNO by extending frequencies into the complex domain, justified by the Ehrenpreis-Palamodov principle for linear PDEs, and then shows big improvements when learning state and optimal control on the nonlinear Burgers' equation. The new part is the explicit link from that integral representation to the inverse Fourier step inside the FNO layer. It is a targeted change rather than a generic deepening of the network. If the reported order-of-magnitude error reductions and better non-periodic boundary predictions hold, that would be useful for control applications. The limitation is that the motivating theorem applies only to linear constant-coefficient equations. Burgers' is nonlinear, so the complex extension lacks a supporting derivation for this case. The gains could come from the added expressivity of complex weights instead of the intended representation. The abstract gives no ablation or parameter-matched comparison to confirm the source of the improvement. This is for people already working with neural operators on PDE control problems. It gives them a concrete variant to try. The work shows clear thinking about the math motivation even if the nonlinear transfer is not fully justified yet. I would send it to peer review. The idea is specific enough that referees can check the experiments and ask for the missing controls.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes FNO$^{∠θ}$, an extension of the Fourier neural operator that replaces the real-frequency inverse Fourier transform in each FNO layer with a complex-frequency version. The change is motivated by the Ehrenpreis-Palamodov fundamental principle, which represents solutions and optimal controls of linear constant-coefficient PDEs as integrals over complex frequencies. The authors apply the resulting architecture to learning both state evolution and linear-quadratic optimal control, and report empirical results on the nonlinear Burgers' equation that show order-of-magnitude reductions in training error and improved accuracy on non-periodic boundaries relative to standard FNO.

Significance. If the observed gains on Burgers' can be shown to arise from the complex-frequency construction rather than from added model capacity, the work would supply a theoretically grounded architectural prior for neural operators on distributed-parameter control problems. The explicit link to the Ehrenpreis-Palamodov principle is a distinctive strength that could improve generalization and interpretability in learning-based PDE control.

major comments (2)

[Abstract and §3] Abstract and §3 (method description): the Ehrenpreis-Palamodov representation is stated only for linear constant-coefficient PDEs, yet the same complex-frequency modification is applied without derivation or analogous representation to the nonlinear Burgers' equation. This leaves the central motivation unsupported for the reported experiments.
[Experiments] Experiments section: no ablation is provided that isolates the complex-frequency extension from the increase in expressivity (complex weights, additional parameters). Without such controls it is impossible to determine whether the order-of-magnitude error reduction is due to the claimed integral representation or simply to greater model capacity.

minor comments (2)

[Title and Abstract] The symbol ∠θ in the title and abstract is never defined; a brief explanation of its relation to the complex-frequency extension would improve clarity.
[Abstract] The abstract refers to 'linear quadratic additive optimal control' without specifying the precise cost functional or the admissible control set; adding one sentence would make the problem statement self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will incorporate revisions to improve clarity and experimental rigor.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (method description): the Ehrenpreis-Palamodov representation is stated only for linear constant-coefficient PDEs, yet the same complex-frequency modification is applied without derivation or analogous representation to the nonlinear Burgers' equation. This leaves the central motivation unsupported for the reported experiments.

Authors: The Ehrenpreis-Palamodov principle is invoked in Section 3 solely to derive the complex-frequency integral representation for linear constant-coefficient PDEs and their optimal controls; the FNO layer modification follows directly from replacing the real-frequency inverse Fourier transform with its complex counterpart to realize that representation. For the nonlinear Burgers' equation we make no claim of an analogous representation and present the results as an empirical test of whether the same architectural change yields practical benefits on a nonlinear distributed-parameter problem. The observed order-of-magnitude error reductions and improved non-periodic boundary predictions are therefore offered as evidence of utility rather than as a theoretical extension. We will revise the abstract and Section 3 to state this distinction explicitly. revision: yes
Referee: [Experiments] Experiments section: no ablation is provided that isolates the complex-frequency extension from the increase in expressivity (complex weights, additional parameters). Without such controls it is impossible to determine whether the order-of-magnitude error reduction is due to the claimed integral representation or simply to greater model capacity.

Authors: We agree that an ablation isolating the complex-frequency mechanism from the added capacity of complex arithmetic is required. The current comparisons use a standard real-valued FNO baseline; the FNO^{∠θ} variant necessarily employs complex weights when frequencies become complex. In the revision we will add two controls: (i) a complex-weighted FNO restricted to real frequencies (by zeroing imaginary parts after each layer or equivalent), and (ii) a real-valued FNO whose number of Fourier modes is increased to match the parameter count of FNO^{∠θ}. Parameter counts for all models will be reported. These additions will allow readers to attribute performance gains more precisely. revision: yes

Circularity Check

0 steps flagged

No circularity: external theorem motivates change; nonlinear results are empirical illustration

full rationale

The paper invokes the Ehrenpreis-Palamodov fundamental principle (an independent external result) to establish an integral representation over complex frequencies for linear constant-coefficient PDEs, then uses that observation to motivate replacing the real-frequency inverse Fourier transform inside the FNO layer with a complex-frequency version. For the nonlinear Burgers' equation the same architectural change is applied and performance is reported via training-error numbers and boundary-value accuracy; no equation or section claims that the complex-integral representation itself holds for the nonlinear operator or derives the observed error reduction from the linear case. No self-citation is load-bearing, no fitted parameter is relabeled as a prediction, and no step equates an output quantity to its input by definition. The derivation chain therefore remains self-contained against external mathematical benchmarks and numerical experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests primarily on the Ehrenpreis-Palamodov principle as an external axiom for the linear case and on empirical results for the nonlinear demonstration; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Ehrenpreis-Palamodov fundamental principle: solutions and optimal controls of linear PDEs with constant coefficients admit integral representations in the complex domain.
Directly invoked to justify extending the frequency variable from real to complex in the FNO inverse Fourier transform.

pith-pipeline@v0.9.0 · 5467 in / 1266 out tokens · 59324 ms · 2026-05-10T19:16:44.488871+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

modify the FNO layer by extending the frequency variable in the inverse Fourier transform from the real to complex domain
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Ehrenpreis-Palamodov fundamental principle... integral of exponential functions... over structured subsets of C^n

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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