In a groundbreaking stride for applied mathematics and artificial intelligence, researchers at the University of Pennsylvania’s School of Engineering and Applied Science have unveiled an innovative method to solve inverse partial differential equations (PDEs) with unprecedented efficiency and stability. This novel approach, termed “Mollifier Layers,” promises to transform how complex scientific problems – from genetic regulation to climate modeling – are approached, by reimagining the mathematical foundations underpinning AI-driven solutions.

Inverse partial differential equations represent one of the most vexing challenges in contemporary science and engineering. Unlike classical PDEs, which forecast system behaviors from known parameters, inverse PDEs operate in reverse: they seek to deduce the hidden parameters or forces that must have existed to produce observed phenomena. This capability is crucial for disciplines where direct measurement of these underlying drivers is impossible, making the inferential power of inverse PDEs indispensable for breakthroughs in areas such as genomics, materials science, and meteorology.

The team, led by Eduardo D. Glandt President’s Distinguished Professor Vivek Shenoy, recognized that solving inverse PDEs is analogous to deducing the precise location where a pebble disturbed a pond simply by observing the resulting pattern of ripples. This metaphor underscores the intrinsic difficulty: the visible effects are clear, yet reconstructing the originating causes requires navigating layers of uncertainty, noise, and computational complexity.

Central to the problem is the mathematical operation of differentiation, which measures how quantities evolve or transform. Higher-order derivatives capture increasingly intricate changes, offering deeper insights into dynamic systems. Traditionally, AI methodologies rely on recursive automatic differentiation within neural networks to compute these derivatives. This process, while powerful, suffers from instability and amplified noise sensitivity when tackling high-order inverse PDEs, particularly with noisy or imperfect data — common in real-world scientific measurements.

The Penn team’s breakthrough was to revisit a classical mathematical concept from the mid-20th century — mollifiers. Introduced by mathematician Kurt Otto Friedrichs in the 1940s, mollifiers are smoothing functions designed to “mollify,” or alleviate, the jaggedness and noise inherent in complex signals before analyzing their behavior. By integrating mollifiers into neural network architectures as specialized “mollifier layers,” the researchers enabled a pre-processing smoothing step that fundamentally enhances the stability and reliability of derivative computations.

This innovation circumvents the pitfalls of recursive automatic differentiation by dampening noise before differentiation occurs, thereby mitigating error amplification and reducing computational demand. The resulting framework not only conserves energy by lowering the scaling of power consumption but also enhances robustness against data imperfections — critical advantages for applications dealing with high-fidelity scientific data.

One particularly compelling application of mollifier layers lies in the field of chromatin biology. Chromatin, the highly organized complex of DNA and proteins within a cell nucleus, regulates gene expression by modulating accessibility to genetic material. Understanding the epigenetic processes that govern chromatin’s folding into tiny domains — mere 100 nanometers across — has challenged scientists for years. Despite advances in microscopy that unveil chromatin’s structure, inferring the chemical reaction rates driving gene regulation remained elusive.

Leveraging mollifier layers, Shenoy’s lab has pioneered a method to reverse-engineer these hidden epigenetic reaction rates from observable chromatin configurations. Such insights could illuminate how genomic accessibility changes over the course of development, aging, and disease — opening avenues for targeted therapies that modulate gene expression by altering these molecular reaction speeds. The implications for personalized medicine and regenerative biology are profound, promising tools to reprogram cells toward desired fates by guided manipulation of chromatin dynamics.

Beyond biology, the versatility of mollifier layers offers transformative potential across scientific machine learning. Fields like materials science and fluid mechanics frequently grapple with noisy, high-dimensional data governed by complex PDEs that are prohibitively difficult to invert reliably. This new method promises to render such inverse problems solvable with greater fidelity and computational efficiency, empowering scientists to extract hidden rules governing diverse natural systems.

The approach marks a significant philosophical shift in scientific AI. Rather than relying solely on brute computational power — a trend exemplified by ever-larger neural networks — this technique underscores the enduring value of mathematical ingenuity. It exemplifies how revisiting foundational mathematical concepts can unlock new capabilities within modern AI frameworks, enhancing their analytical reach while reducing resource consumption.

As the research team prepares to present their findings at the Conference on Neural Information Processing Systems (NeurIPS 2026) and publish in Transactions on Machine Learning Research, the scientific community anticipates wide adoption of mollifier layers. These layers elegantly bridge mathematics and AI, transforming noisy real-world data into actionable scientific insight.

Ultimately, the power to infer the hidden causes behind complex observable patterns empowers researchers not only to understand but to alter the dynamics of physical, biological, and engineered systems. The advent of mollifier layers thus heralds a new era of scientific discovery, where understanding and control are unlocked through mathematically grounded, computationally efficient AI.

By advancing from mere observation toward comprehensive causality reconstruction, this mathematical innovation advances humanity’s ability to decode the complexities of nature — from the microscopic choreography of DNA inside cells to the vast, turbulent currents of Earth’s atmosphere.

Subject of Research: Not applicable

Article Title: Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning

News Publication Date: 9-Mar-2026

References:
– Transactions on Machine Learning Research (TMLR), openreview.net/forum?id=6mFVZSzyev
– Prior work on chromatin organization, Nature Communications, DOI link: 10.1038/s41467-026-71213-5
– Kurt Otto Friedrichs’ original mollifier paper, Transactions of the American Mathematical Society, 1944

Image Credits: Sylvia Zhang, Penn Engineering

Keywords: inverse partial differential equations, mollifier layers, neural networks, scientific machine learning, chromatin biology, epigenetics, high-order derivatives, AI stability, computational efficiency, PDE inversion, gene regulation, mathematical smoothing

At the University of Waterloo, a pioneering research group is tackling this challenge by blending the rigor of applied mathematics with the capabilities of modern machine learning. Their work focuses on providing mathematically sound verification for AI systems that govern time-varying physical processes. Dr. Jun Liu, a professor of applied mathematics and Canada Research Chair in Hybrid Systems and Control, leads this effort by leveraging classical tools such as differential equations to model dynamic systems characterized by continuous change, from the intricate flow of electricity across power grids to the nuanced movements of autonomous vehicles.

Central to their approach is the deployment of Lyapunov functions, a mathematical concept dating back to the late 19th century, which serve as certificates of stability for complex dynamical systems. If one imagines the behavior of a system as analogous to a ball rolling within a landscape, a Lyapunov function acts like the contours of a bowl that ensure the ball eventually settles into a stable equilibrium at the bottom. However, finding an appropriate Lyapunov function for modern, nonlinear systems controlled by AI has historically been a formidable mathematical problem, often requiring manual derivation and deep expertise.

To circumvent this bottleneck, Liu’s team harnessed the power of neural networks, a subclass of AI increasingly recognized for their ability to approximate complex functions and solve intricate problems. The innovation lies in training a neural network to satisfy the stringent mathematical constraints that define a valid Lyapunov function. More specifically, the network learns a function that confirms whether the system’s state will converge to a safe, stable operating point over time. Training such networks involves embedding the underlying physics and control laws of the system into the learning process itself—a method the researchers refer to as physics-informed neural networks.

However, the use of neural networks in safety-critical control systems introduces new questions about verification. Standard AI models can be opaque, leading to a lack of trust in their outputs. To address this, the Waterloo team supplemented their approach with a separate logic-driven reasoning system, a form of AI specializing in formal verification techniques. This reasoning layer rigorously checks whether the neural network’s output adheres to the mathematical criteria of safety, creating a closed loop of learning and verification that collectively provides a rare mathematical guarantee of system stability.

One of the remarkable outcomes of this dual AI approach—using one AI to design controllers and another to verify them—is the substantial reduction in the traditionally labor-intensive process of controller design and safety proof construction. What once required painstaking manual derivation and exhaustive analytical effort can now be performed more efficiently without compromising on the level of rigor demanded by critical applications.

Importantly, this approach does not advocate for the replacement of humans in the decision-making loop but rather for augmenting human capabilities. Dr. Liu emphasizes that ethical considerations and higher-order judgments remain the province of human experts. The AI systems are designed to offload the computationally demanding and error-prone tasks of mathematical proof generation and real-time control optimization. This symbiosis enables researchers and engineers to concentrate on oversight, interpretation, and policy, domains that intrinsically require human values and intuition.

To validate their framework, the researchers applied their combined machine learning and formal verification toolbox to several challenging control scenarios. These case studies demonstrated that their method either matched or outperformed classical techniques in ensuring system safety and stability. By incorporating physics knowledge directly into the learning architecture, they achieved more reliable and interpretable results than conventional black-box machine learning controllers could offer.

Looking forward, the team is actively developing their framework into an open-source software toolbox. This initiative promises to democratize access to advanced verification tools for the broader scientific and engineering communities, accelerating innovation in safe AI control applications. Furthermore, collaborations with industry partners are underway, aiming to translate these theoretical advances into practical, high-impact solutions for sectors where AI safety is paramount.

This research aligns closely with global efforts to foster transparent, responsible, and trustworthy AI technologies. At Waterloo, the project benefits from synergies with initiatives such as the TRuST Scholarly Network, which fosters interdisciplinary research to ensure AI systems adhere to ethical and safety standards. Concurrently, federal programs aimed at promoting accountable AI underscore the timeliness and societal importance of this work.

The technical breakthrough reported in the study, “Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification,” published in the journal Automatica, reflects a significant step toward integrating rigorous mathematical theory with state-of-the-art AI methods. By bridging these domains, the research outlines a promising pathway to endow AI-driven systems with verifiable safety properties, thus bolstering confidence in their deployment in real-world applications.

In summary, the University of Waterloo team’s novel approach represents a paradigm shift in ensuring the safe operation of AI controllers in dynamic physical systems. By combining the strengths of neural networks in function approximation with formal logic-based verification, they provide a comprehensive framework that addresses the long-standing challenge of guaranteeing AI safety. Their work underscores the potential of interdisciplinary innovation to solve some of the most pressing technological problems of our age, paving the way for a future in which AI-driven systems can be both powerful and trustworthy.

Subject of Research: Safe and trustworthy AI for dynamic physical systems through mathematical verification and machine learning

Article Title: Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification

News Publication Date: Not specified

Web References:

• https://uwaterloo.ca/applied-mathematics/profiles/jun-liu
• https://uwaterloo.ca/trust-research-undertaken-science-technology-scholarly-network/
• https://www.sciencedirect.com/science/article/pii/S000510982500086X

References:

• Liu, J. et al. (2025). Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification. Automatica.

Keywords:
Applied mathematics, Machine learning, Autonomous vehicles, Autonomous robots, Artificial neural networks, Neural networks, Applied sciences and engineering, Systems theory, Adaptive systems