The team’s solution, called “Mollifier Layers,” improves how AI handles these problems by refining the math behind the process instead of simply increasing computing power. The approach could have wide-ranging applications, from decoding genetic activity to improving weather predictions.

“Solving an inverse problem is like looking at ripples in a pond and working backward to figure out where the pebble fell,” says Vivek Shenoy, Eduardo D. Glandt President’s Distinguished Professor in Materials Science and Engineering (MSE) and senior author of a study published in Transactions on Machine Learning Research (TMLR), which will be presented at the Conference on Neural Information Processing Systems (NeurIPS 2026). “You can see the effects clearly, but the real challenge is inferring the hidden cause.”

Instead of relying on more powerful hardware, the researchers focused on improving the underlying mathematics. “Modern AI often advances by scaling up computation,” says Vinayak Vinayak, a doctoral candidate in MSE and co-first author of the study. “But some scientific challenges require better mathematics, not just more compute.”

Why Inverse PDEs Matter in Science

Differential equations are the backbone of scientific modeling. They describe how systems change over time, whether it is population growth, heat flow, or chemical reactions.

Partial differential equations extend this idea further by capturing how systems evolve across both space and time. Scientists use them to study everything from weather patterns to how heat moves through materials and even how DNA is organized inside cells.

Inverse PDEs go a step further. Rather than predicting outcomes based on known rules, they allow scientists to start with observed data and work backward to uncover the hidden forces driving those observations.

“For years, we’ve used these equations to study how chromatin, which is the folded state of DNA inside the nucleus, organizes itself inside living cells,” says Shenoy. “But we kept running into the same problem: We could see the structures and model their formation, but we could not reliably infer the epigenetic processes driving this system, namely the chemical changes that help control which genes are active. The more we tried to optimize the existing approach, the clearer it became that the mathematics itself needed to change.”

Rethinking How AI Handles Complex Math

A key concept behind these equations is differentiation, which measures how something changes. Simple derivatives show how fast something increases or decreases, while higher-order derivatives capture more intricate patterns.

Traditionally, AI systems compute these derivatives using a process called recursive automatic differentiation. This method repeatedly calculates changes as data moves through a neural network, the foundation of modern AI.

However, this approach struggles when dealing with complex systems and noisy data. It can become unstable and demand enormous computing resources.

The researchers compare it to repeatedly zooming in on a rough, jagged line. Each step amplifies imperfections, making the final result less reliable. To overcome this, the team realized they needed a way to smooth the data before analyzing it.

Mollifier Layers Offer a Smarter Solution

The answer came from a concept introduced in the 1940s by mathematician Kurt Otto Friedrichs, who described “mollifiers,” tools designed to smooth irregular or noisy functions.

By adapting this idea, the researchers created a “mollifier layer” within AI models. This layer smooths the input data before calculating changes, avoiding the instability caused by traditional methods.

“We initially assumed the issue had to do with neural network’s architecture,” says Ananyae Kumar Bhartari, a graduate of Penn Engineering’s Scientific Computing master’s program and the paper’s other co-first author. “But, after carefully adjusting the network, we eventually realized the bottleneck was recursive automatic differentiation itself.”

The results were striking. The new method reduced noise and significantly lowered the computational cost required to solve these equations.

Implementing a “mollifier layer,” which smoothed the signal before measuring it, radically diminished both the noisiness and the power consumption scaling. “That let us solve these equations more reliably, without the same computational burden,” says Bhartari.

Unlocking the Secrets of DNA Organization

One of the most promising applications of this approach lies in understanding chromatin, the complex structure of DNA and proteins inside cells.

These structures operate at an incredibly small scale, but they play a major role in determining how genes are turned on or off.

“These domains are just 100 nanometers in size,” says Shenoy, “but because accessibility determines gene expression, and gene expression governs cell identity, function, aging and disease, these domains play a critical role in biology and health.”

By estimating the rates of epigenetic reactions, which control gene activity, the new AI method could help scientists move beyond simply observing chromatin to predicting how it changes over time.

“If we can track how these reaction rates evolve during aging, cancer or development,” adds Vinayak, “this creates the potential for new therapies: If reaction rates control chromatin organization and cell fate, then altering those rates could redirect cells to desired states.”

Beyond Biology: Wide-Ranging Scientific Impact

The potential uses of mollifier layers extend far beyond genetics. Many areas of science, including materials research and fluid dynamics, involve complex equations and noisy data.

This new framework could provide a more stable and efficient way to uncover hidden parameters across a wide variety of systems.

The researchers see this as a step toward a larger goal: turning observations into deeper understanding.

“Ultimately, the goal is to move from observing complex patterns to quantitatively uncovering the rules that generate them,” says Shenoy. “If you understand the rules that govern a system, you now have the possibility of changing it.”

This study was conducted at the University of Pennsylvania School of Engineering and Applied Science and supported by National Cancer Institute (NCI) Award U54CA261694 (V.B.S.); National Science Foundation (NSF) Center for Engineering Mechanobiology (CEMB) Grant CMMI -154857 (V.B.S.); NSF Grant DMS -2347834 (V.B.S.); National Institute of Biomedical Imaging and Bioengineering (NIBIB) Awards R01EB017753 (V.B.S) and R01EB030876 (V.B.S.) and National Institute of General Medical Sciences (NIGMS) Award R01GM155943 (V.B.S).

The trial, led by researchers at UCL and UCLH, found that just nine weeks of treatment with pembrolizumab before surgery led to strong and lasting responses in patients with stage two or three colorectal cancer.

No Cancer Recurrence After Nearly Three Years

Early results showed that 59% of patients had no detectable cancer after completing immunotherapy and undergoing surgery. Now, after 33 months of follow-up, none of the patients have experienced a relapse.

This includes both patients whose tumors completely disappeared and those who still had small traces of cancer after treatment. In all cases, the remaining cancer did not grow or spread over time.

This outcome stands in contrast to standard care, where about 25% of patients treated with surgery followed by chemotherapy are expected to see their cancer return within three years. The findings suggest that starting with immunotherapy may offer longer-lasting protection.

Personalized Blood Tests May Predict Treatment Success

Researchers also explored why the treatment worked so well and how to identify patients most likely to benefit. By analyzing blood samples, they developed personalized tests that can detect whether cancer DNA is still present in the bloodstream.

These tests may allow doctors to determine early on whether the treatment has been successful.

Dr. Kai-Keen Shiu, Chief Investigator of the trial from UCL Cancer Institute and a Consultant Medical Oncologist at UCLH, said: “Seeing that no patients have experienced a cancer recurrence after almost three years of follow-up is extremely encouraging and strengthens our confidence that pembrolizumab is a safe and highly effective treatment to improve outcomes in patients with high-risk bowel cancers.

“What is particularly exciting is that we now may be able to predict who will respond to the treatment using personalized blood tests and immune profiling. These tools could help us tailor our approach, identifying patients who are doing well and may need less therapy before and after surgery versus patients at higher risk of disease progression or relapse who need additional treatment.”

Understanding Colon Cancer Risk and Survival

Bowel cancer is the fourth most common cancer in the UK, with about 44,000 new cases each year. While it primarily affects older adults, diagnoses among people under 50 have been rising.

Outcomes depend heavily on how early the cancer is detected. Around 90% of patients with stage one bowel cancer survive at least five years. Survival drops to 65% at stage three and just 10% at stage four. Some tumor types are also more likely to resist treatment and return.

Trial Details and Patient Group

The NEOPRISM-CRC trial included 32 patients with stage two or three colorectal cancer and a specific genetic subtype (MMR deficient/MSI-high bowel cancer). This subtype accounts for about 10-15% of such cases, or roughly 2,000-3,000 patients each year in the UK.

Participants received up to nine weeks of pembrolizumab before undergoing surgery, instead of the usual approach of surgery followed by several months of chemotherapy. They were then monitored over time.

The latest findings were presented at the American Association for Cancer Research (AACR) Annual Meeting 2026 in San Diego. The study involved multiple UK hospitals, with UCL and UCLH leading the research and biotech company Personalis contributing to the analysis.

Why Immunotherapy Works So Well

Scientists also gained new insights into how immunotherapy produces these lasting effects.

Professor Marnix Jansen from UCL Cancer Institute and UCLH said: “These results not only confirm the durability of responses we saw almost three years ago, but also provide crucial biological insights into why immunotherapy is so effective in this setting.”

Researchers found that when tumor DNA disappeared from the blood, patients were far more likely to remain cancer-free long term.

Yanrong Jiang, first author of the study, said: “As a research team, we were thrilled to be able to follow patients very closely using the personalized blood tests. When tumor DNA disappeared from the blood, patients were much more likely to have no cancer remaining, and this matched the long-term results we’re now seeing.

“In addition, we also saw that immune profiling from tumor tissue, before patients start their first cycle of treatment, can help to predict response. We hope these tests may be used to guide treatment decisions in a more practical and timely way.”

Patient Story Highlights Real-World Impact

Christopher Burston, a 73-year-old patient from Portland, Dorset, was diagnosed with colorectal cancer in February 2023 after routine screening detected blood in his stool.

He said: “One came back with indications of blood in my stool. I went through further tests, and it was at the colonoscopy that they identified a cancer in my bowel.”

Soon after his diagnosis, he was offered the chance to join the NEOPRISM trial and chose to participate, traveling to London for treatment.

He received three doses of immunotherapy over nine weeks before undergoing surgery in May 2023. His recovery was smooth, with minimal side effects.

He said: “The outcome of the surgery was essentially that the cancer had melted away, these were the doctor’s words. The immunotherapy had had an almost immediate effect. I saw the images when I had the first colonoscopy and could see it was really quite a substantial lump. So as I say, it wasn’t a minor thing, I was diagnosed with stage 3 cancer.”

Nearly three years later, he remains cancer-free and has returned to his normal routine.

Christopher said: “The recovery went fine. I didn’t have any problems. And since then, I’ve been feeling pretty much back to normal. I feel very lucky that I’ve reached the stage where my main problem is age rather than cancer or any illness.”

Partial meniscectomy is one of the most common orthopedic surgeries worldwide. While its use has declined in Finland in recent years, it remains a routine treatment in many countries.

10-Year Study Finds Worse Outcomes After Surgery

The long-term results paint a concerning picture. Patients who underwent partial meniscectomy did not experience better outcomes than those who had sham surgery. In fact, they tended to do worse.

After a decade, these patients reported more knee symptoms and poorer function. They also showed greater progression of osteoarthritis and were more likely to need additional knee surgery compared to those who did not receive the actual procedure.

Unique Trial Design Strengthens Findings

The Finnish Degenerative Meniscal Lesion Study (FIDELITY) stands out for its rigorous design. It included a sham surgery control group, allowing researchers to directly compare outcomes against a placebo procedure. Participants with degenerative meniscal tears were randomly assigned to receive either partial meniscectomy or sham surgery, and their progress was tracked for 10 years.

Teppo Järvinen, Professor at the University of Helsinki and the principal investigator of the FIDELITY emphasizes the broader significance of the results:

“Our findings suggest that this may be an example of what is known as a medical reversal, where broadly used therapy proves ineffective or even harmful.”

Rethinking the Cause of Knee Pain

The surgery has long been based on the idea that knee pain, especially on the inner side, is caused by a meniscus tear that can be fixed surgically. However, this assumption may not hold up.

“The surgery is based on the assumption that pain in the inside of the knee is caused by a medial meniscus tear, which can be treated surgically. Such reasoning — assumption based on biological credibility — is still very common in medicine but in this case, the assumption does not withstand critical examination. Based on current understanding, pain in various joints, such as the knee joint in this case, is related to degeneration brought about by aging,” says Raine Sihvonen, Specialist in Orthopaedics and Traumatology and the other principal investigator of the FIDELITY study.

Concerns About Risks and Long-Term Harm

Earlier registry and observational studies have already raised red flags about potential downsides of this surgery. These include a higher likelihood of arthroplasty, or joint replacement surgery, and a possible increase in complications after the procedure. However, observational data alone cannot prove cause and effect.

“Several randomized studies have already demonstrated that partial meniscectomy has not improved patients’ symptoms or function in the short (1-2 years) or medium (5 years) term. Regardless, the procedure has remained widely used in many countries,” says Doctoral Researcher and Specialist in Orthopaedics and Traumatology, Dr. Roope Kalske.

Why the Procedure Is Still Widely Used

Despite mounting evidence, changing clinical practice has been slow.

“For nearly a decade, many independent, non-orthopedic organizations providing clinical guidelines have recommended that the procedure should be discontinued. Still, for example, the American Academy of Orthopedic Surgeons (AAOS) and the British Association for Surgery of the Knee (BASK) have continued to endorse the surgery.

“This effectively illustrates how difficult it is to give up inefficient therapies,” Järvinen sums up.

Strong Collaboration Behind the Study

The research was carried out across five hospitals, highlighting strong collaboration and patient commitment. Of the original 146 participants, more than 90% completed the final follow-up phase.

“The study conducted in five hospitals is an example of smooth multicenter collaboration, as well as the commitment of research patients to an interesting project. Of the original 146 participants, more than 90% took part in the final stage of the study,” says the research manager Pirjo Toivonen.

The Finnish Degenerative Meniscal Lesion Study FIDELITY) is part of the broader work of the FICEBO research group in assessing the impact of surgical therapies. The project is a collaboration between the university hospitals of Helsinki, Kuopio and Turku, Hatanpää Hospital in Tampere, Hospital Nova in Jyväskylä and the Finnish Institute for Health and Welfare.