Fast training of accurate physics-informed neural networks without gradient descent

Abhishek Chandra; Anna Veselovska; Chinmay Datar; Erik Lien Bolager; Felix Dietrich; Iryna Burak; Massimo Fornasier; Qing Sun; Taniya Kapoor

arxiv: 2405.20836 · v3 · submitted 2024-05-31 · 🧮 math.NA · cs.CE· cs.LG· cs.NA

Fast training of accurate physics-informed neural networks without gradient descent

Chinmay Datar , Taniya Kapoor , Abhishek Chandra , Qing Sun , Erik Lien Bolager , Iryna Burak , Anna Veselovska , Massimo Fornasier

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Felix Dietrich

This is my paper

Pith reviewed 2026-05-24 00:47 UTC · model grok-4.3

classification 🧮 math.NA cs.CEcs.LGcs.NA

keywords physics-informed neural networksrandom featuresspace-time separationPDE solversgradient-free trainingtemporal causality

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The pith

Frozen-PINNs solve time-dependent PDEs by replacing gradient descent with a linear solve over random features and explicit space-time separation.

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

The paper presents Frozen-PINN as a way to approximate solutions to time-dependent partial differential equations without relying on iterative gradient-based optimization. It separates space and time explicitly so that temporal causality is enforced by construction, then uses random features to reduce the problem to a single linear solve for the coefficients. This approach is tested on eight benchmarks that include extreme advection, shocks, and high-dimensional cases, where it reports faster training and higher accuracy than existing PINN methods. A reader would care because the usual barriers of slow, unstable training and non-causal time handling have limited practical use of PINNs in science and engineering. The work therefore questions whether stochastic gradient descent is necessary for accurate physics-informed neural network solutions.

Core claim

Frozen-PINN leverages the principle of space-time separation that uses random features instead of training with gradient descent, and incorporates temporal causality by construction. On eight PDE benchmarks, including challenges such as extreme advection speeds, shocks, and high dimensionality, Frozen-PINNs achieve superior training efficiency and accuracy over state-of-the-art PINNs, often by several orders of magnitude. The method addresses longstanding training and accuracy bottlenecks of PINNs, delivering quickly trainable, highly accurate, and inherently causal PDE solvers.

What carries the argument

Frozen-PINN, which approximates the solution via random spatial features whose coefficients are obtained from a linear solve after enforcing explicit space-time separation to guarantee causality.

If this is right

PINN training time drops from many hours of gradient descent to a single linear solve on the tested benchmarks.
Temporal causality holds automatically, removing the need for separate causal weighting or time-marching schemes.
The reliance on specialized hardware for large-scale stochastic gradient descent is no longer required for these PDE problems.
A new reference point is established for comparing future PINN variants on the same set of challenging time-dependent equations.

Where Pith is reading between the lines

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

The same random-feature-plus-linear-solve pattern could be tested on other families of differential equations that currently rely on gradient-based neural solvers.
If the linear solve remains stable at higher dimensions, the approach may reduce the memory and compute scaling problems that iterative methods face in three or more spatial variables.
One could examine whether the fixed random-feature basis limits expressivity on problems whose solutions contain sharp moving fronts not captured by the initial feature set.

Load-bearing premise

A linear solve over random features combined with explicit space-time separation is sufficient to produce accurate solutions to the target time-dependent PDEs without any iterative optimization.

What would settle it

Running the eight reported benchmarks with the same random-feature basis and observing that Frozen-PINN does not match or exceed the accuracy and speed of the compared state-of-the-art PINNs on at least one case with shocks or extreme advection.

read the original abstract

Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their accuracy and training speed are limited by two core barriers: gradient-descent-based iterative optimization over complex loss landscapes and non-causal treatment of time as an extra spatial dimension. We present Frozen-PINN, a novel PINN based on the principle of space-time separation that leverages random features instead of training with gradient descent, and incorporates temporal causality by construction. On eight PDE benchmarks, including challenges such as extreme advection speeds, shocks, and high dimensionality, Frozen-PINNs achieve superior training efficiency and accuracy over state-of-the-art PINNs, often by several orders of magnitude. Our work addresses longstanding training and accuracy bottlenecks of PINNs, delivering quickly trainable, highly accurate, and inherently causal PDE solvers, a combination that prior methods could not realize. Our approach challenges the reliance of PINNs on stochastic gradient-descent-based methods and specialized hardware, leading to a paradigm shift in PINN training and providing a challenging benchmark for the community.

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

Frozen-PINN swaps gradient descent for a random-feature linear solve plus explicit space-time split to enforce causality, with large reported gains on eight benchmarks that still need verification on residual control for nonlinear cases.

read the letter

The main claim is that Frozen-PINN uses random features for the spatial part and solves a linear system for the coefficients, with time handled by an explicit separation that enforces causality by construction. This avoids any gradient descent. What is new is the specific way they combine random features with the space-time split to get both speed and causality without optimization. On the eight benchmarks they show faster training and better accuracy than existing PINNs, sometimes by orders of magnitude. This is useful because it directly tackles the training speed issue that has held back PINNs for time-dependent problems. If the linear solve works reliably, it could make these methods more practical. The soft spot is in the nonlinear and shock cases. A linear solve over random features does not automatically make the PDE residual small when the equation has nonlinear terms or discontinuities. The paper would need to show that the chosen features and any regularization do not amount to hidden tuning, and that the residual is actually controlled. The abstract gives no equations, so the full text needs to make the setup transparent. This paper is for people working on neural network methods for PDEs who are looking for training-free or training-light alternatives. A reader who wants to test whether random feature methods can replace optimized PINNs would get value from the benchmarks. It deserves peer review because the claims are specific and the method is testable, even if revisions will likely be needed for the details.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Frozen-PINN, a physics-informed neural network for time-dependent PDEs that employs random features with explicit space-time separation, trains via a single linear solve instead of gradient descent, and enforces temporal causality by construction. It reports superior training efficiency and accuracy, often by orders of magnitude, over state-of-the-art PINNs on eight benchmarks that include extreme advection speeds, shocks, and high-dimensional problems.

Significance. If the central claims hold, the work would be significant for removing iterative optimization and non-causal time treatment from PINN training, delivering fast, accurate, inherently causal solvers and providing a new benchmark that challenges reliance on SGD-based methods.

major comments (2)

[Abstract] Abstract: the claim of 'superior training efficiency and accuracy ... often by several orders of magnitude' on eight benchmarks is unsupported by any equations, error metrics, baseline comparisons, or implementation details, rendering the data unverifiable against the stated performance.
[Method] Method (space-time separation and random-feature linear solve): the premise that a single least-squares solve over random spatial features automatically yields a small PDE residual for nonlinear terms or shocks is not automatic; without a proof or ablation showing that the chosen feature distribution and regularization suffice without implicit compensation, the 'no iterative optimization' claim rests on an unverified assumption.

minor comments (1)

The abstract and title could more precisely indicate the class of PDEs addressed and the precise form of the random-feature basis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive comments. We address each major comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of 'superior training efficiency and accuracy ... often by several orders of magnitude' on eight benchmarks is unsupported by any equations, error metrics, baseline comparisons, or implementation details, rendering the data unverifiable against the stated performance.

Authors: Abstracts are concise summaries and are not expected to contain equations, full metrics, or implementation details. The manuscript provides these in the numerical experiments section, including tables with quantitative error metrics (e.g., L2 errors), training times, baseline comparisons against state-of-the-art PINNs, and implementation details for all eight benchmarks, which substantiate the orders-of-magnitude claims. revision: no
Referee: [Method] Method (space-time separation and random-feature linear solve): the premise that a single least-squares solve over random spatial features automatically yields a small PDE residual for nonlinear terms or shocks is not automatic; without a proof or ablation showing that the chosen feature distribution and regularization suffice without implicit compensation, the 'no iterative optimization' claim rests on an unverified assumption.

Authors: The space-time separation reduces the problem to a linear system in the random feature coefficients whose solution minimizes the PDE residual in the least-squares sense over the chosen basis. For nonlinear terms and shocks the method depends on the approximation power of the random features together with the explicit causality constraint; the eight benchmarks include nonlinear and shock cases where the resulting residuals are small. We acknowledge that a general theoretical guarantee is not provided. We will add an ablation study on feature distribution and regularization parameters in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity; method is a proposed algorithmic construction

full rationale

The paper introduces Frozen-PINN as a new procedure that replaces gradient-descent training with a linear solve over random features plus explicit space-time separation. No equations or fitting steps are exhibited in the provided abstract that reduce a claimed prediction back to its own inputs by construction. The central claim (superior accuracy and speed on benchmarks) rests on the empirical performance of the described algorithm rather than on any self-definitional loop, fitted-input-renamed-as-prediction, or load-bearing self-citation chain. The derivation chain is therefore self-contained as an independent proposal; no load-bearing step collapses to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all such elements remain unknown.

pith-pipeline@v0.9.0 · 5768 in / 1039 out tokens · 25301 ms · 2026-05-24T00:47:49.632264+00:00 · methodology

discussion (0)

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