Penn Engineers Publish Mollifier Layers in TMLR, Replacing Autodiff With Convolutional Smoothing for Inverse PDE Learning
A University of Pennsylvania team led by Vivek Shenoy has published Mollifier Layers in TMLR, a module that swaps recursive automatic differentiation for convolutional smoothing to make physics-informed neural networks stable on high-order, noisy inverse PDEs.
Overview
A team at the University of Pennsylvania School of Engineering and Applied Science has published a new technique for solving inverse partial differential equations with neural networks, called Mollifier Layers, in Transactions on Machine Learning Research (TMLR). The module replaces the recursive automatic differentiation that most physics-informed neural networks rely on with a single convolutional operation built around a classical mathematical object — the mollifier — and the work will be presented at the Conference on Neural Information Processing Systems (NeurIPS 2026), according to Phys.org and EurekAlert!.
Inverse PDE problems ask a model to recover hidden parameters of a physical system from observations of its behavior — the kind of work that underpins materials science, fluid dynamics, weather forecasting and parts of computational biology. The paper, titled “Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning,” is authored by Ananyae Kumar Bhartari, Vinayak Vinayak and Vivek B. Shenoy, per the arXiv preprint.
What the technique does
Most physics-informed neural networks compute the derivatives that appear in a PDE by threading recursive automatic differentiation through the network, repeatedly differentiating the model’s outputs with respect to its inputs. For high-order or noisy systems the procedure can become unstable and memory-hungry. Mollifier Layers sidestep that loop by sitting at the output and applying a single smoothing operation derived from a class of analytic functions known as mollifiers — bell-shaped kernels that were introduced in the 1940s by mathematician Kurt Otto Friedrichs to regularize irregular signals, as ScienceDaily explains.
The arXiv abstract describes the module as “a lightweight, architecture-agnostic module that replaces autodiff with convolutional operations using analytically defined mollifiers,” and reports gains in memory efficiency, training speed and accuracy for parameter recovery across “first-, second-, and fourth-order PDEs — including Langevin dynamics, heat diffusion, and reaction-diffusion systems,” per the arXiv preprint.
Doctoral candidate Ananyae Kumar Bhartari, a co-first author and a graduate of Penn Engineering’s Scientific Computing master’s program, told Phys.org the team’s initial diagnosis of the problem was wrong: “We initially assumed the issue had to do with neural network’s architecture. But we eventually realized the bottleneck was recursive automatic differentiation itself,” according to Phys.org.
What the paper applies it to
The paper’s headline biomedical application is inferring “spatially varying epigenetic reaction rates from super-resolution chromatin imaging data,” per the arXiv preprint. Chromatin — the packed DNA-and-protein structure inside cell nuclei — organizes into compact domains roughly 100 nanometers across, according to Phys.org, and tracking the reaction rates that drive their formation requires inferring rate parameters from noisy microscopy images. Shenoy framed why the existing toolkit fell short: “We could see the structures and model their formation, but we could not reliably infer the epigenetic processes driving this system,” according to Phys.org.
Vivek Shenoy, the Eduardo D. Glandt President’s Distinguished Professor in Materials Science and Engineering and the senior author of the work, described the broader class of problems in plain terms: “Solving an inverse problem is like looking at ripples in a pond and working backward to figure out where the pebble fell,” as ScienceDaily reports.
The Penn team’s framing in the press cycle leans on a methodological argument that runs against the prevailing scaling story in AI. Vinayak Vinayak, a doctoral candidate in Materials Science and Engineering and co-first author, put it this way: “Modern AI often advances by scaling up computation. But some scientific challenges require better mathematics, not just more compute,” according to SciTechDaily.
What the press releases claim about performance
The outlets covering the release describe the gains qualitatively rather than with specific numbers. ScienceDaily characterizes the result as “reduced noise and significantly lowered the computational cost required to solve these equations,” and adds that implementing the smoothing step “radically diminished both the noisiness and the power consumption scaling,” per ScienceDaily. Bhartari summarized the practical effect by saying the module “let us solve these equations more reliably, without the same computational burden,” according to EurekAlert!.
The arXiv abstract claims gains across “memory efficiency, training time and accuracy for parameter recovery,” per the arXiv preprint, but the underlying paper — rather than any of the press releases — is the place to look for benchmark tables. None of the cited press outlets quotes a specific multiplier for the speedup or memory reduction.
What we don’t know
The TMLR publication has not yet been paired with an accompanying open-source release in any of the cited press materials, so independent reproduction will depend on whatever code the authors choose to ship alongside the camera-ready version at NeurIPS 2026. The framing in the press cycle is also narrow: the only worked biomedical application discussed in detail is chromatin reaction-rate inference, and broader claims about applicability to weather, materials science and fluid mechanics rest on the authors’ own characterization rather than on third-party benchmarks.
None of the cited outlets indicates whether Mollifier Layers will be integrated into the major physics-informed neural-network libraries used by other research groups. That question — and how the technique fares against the strongest baselines on standard inverse-PDE benchmarks — will likely be answered at NeurIPS 2026 in December rather than in the press cycle around the TMLR paper.