Shows 40% Data Efficiency Gains with Decoupled Diffusion Sampling for PDEs

Researchers are tackling the challenging problem of solving inverse partial differential equations (PDEs) with limited data, a common issue in fields like medical imaging and materials science. Thomas Y.L. Lin from the University of Washington, Seattle, alongside Jiachen Yao and Lufang Chiang from the California Institute of Technology and National Taiwan University respectively, with Julius Berner and Anima Anandkumar et al. from NVIDIA Corporation and Caltech, present a novel approach called Decoupled Diffusion Sampling (DDIS). This work distinguishes itself by explicitly separating the learning of coefficient priors from the forward PDE solution, offering significant improvements in data efficiency and accuracy compared to existing methods. Their findings, supported by both theoretical proofs and empirical results demonstrating an average error reduction of 11% and spectral error reduction of 54% under sparse observation, represent a substantial advance in physics-informed machine learning and could unlock solutions to previously intractable inverse problems.

Decoupled diffusion models enhance data efficiency in partial differential equation inverse problems by learning separate forward and reverse processes

DDIS separates the learning of the coefficient prior from explicit modeling of the forward PDE, achieving superior data efficiency and effective physics-informed learning. Theoretically, the team proves that DDIS avoids the guidance attenuation failure observed in joint models when training data is limited. This addresses a critical challenge in high-dimensional spaces where sparse data makes effective coefficient-space guidance difficult to achieve.
Empirical evaluations demonstrate that DDIS achieves state-of-the-art performance with sparse observations, improving the l2 error by 11% and spectral error by 54% on average. Notably, when training data is restricted to just 1%, DDIS maintains accuracy with a 40% reduction in l2 error compared to joint models.

The research establishes a new benchmark for inverse PDE solving, particularly in scenarios with limited paired data, such as weather forecasting and geophysical imaging where sensor coverage is often sparse. This work opens avenues for more robust and accurate solutions to ill-posed problems across various scientific and engineering disciplines.

By explicitly representing the underlying physics through a neural operator, DDIS offers a significant advantage over methods that implicitly learn physics from statistical correlations. The decoupling strategy not only enhances data efficiency but also enables the effective use of advanced sampling techniques like DAPS, paving the way for more reliable and high-resolution reconstructions of unknown coefficient fields from limited observations.

Decoupled Diffusion and Neural Operators for Sparse Inverse PDE Solutions offer improved efficiency and accuracy

Scientists developed a Decoupled Inverse Solver (DDIS) to address data efficiency in inverse partial differential equation (PDE) problems. The study pioneered a decoupled design, training an unconditional diffusion model to learn the coefficient prior and employing a neural operator to explicitly model the forward PDE for guidance.

This decoupling enables superior data efficiency and facilitates Decoupled Annealing Posterior Sampling (DAPS) to mitigate over-smoothing issues common in Posterior Sampling (DPS). Researchers rigorously proved that DDIS avoids guidance attenuation failures observed in joint coefficient-solution models when training data is limited.

Experiments employed sparse observation data, where observations were available on only a small fraction of the spatial domain, simulating realistic sensor coverage scenarios. The team trained the neural operator, Lφ(a), using limited paired coefficient-solution data to directly learn the forward physics map, while leveraging abundant unpaired coefficients for prior modelling.

The study harnessed DAPS during inference, utilising the neural operator to propagate sparse observations in the solution space and generate dense guidance over the coefficient space. This approach overcomes the limitations of joint-embedding models, which struggle with sparse guidance signals and subsequent over-smoothing.

DDIS achieved state-of-the-art performance, improving the l2 error by 11% and spectral error by 54% on average compared to existing methods. Notably, when limited to just 1% of the data, DDIS maintained accuracy with a 40% reduction in error relative to joint models. This demonstrates the effectiveness of the decoupled design in leveraging physics consistency and achieving robust performance under extreme data scarcity.

Decoupled diffusion models enhance sparse data solutions for inverse partial differential equations by improving reconstruction accuracy

Scientists have developed a Decoupled Inverse Solver (DDIS), a novel physics-aware generative framework for inverse partial differential equation (PDE) problems, achieving state-of-the-art performance with limited data. The research team decoupled the learning process, training an unconditional diffusion model for the coefficient prior and a neural operator to explicitly model the forward PDE, enabling superior data efficiency.

Experiments revealed that DDIS improves average l2 error by 11% and spectral error by 54% under sparse observation conditions. The core of the breakthrough lies in the Decoupled Annealing Posterior Sampling (DAPS) method, which avoids over-smoothing issues present in traditional Posterior Sampling (DPS) techniques.

Theoretical analysis proves that DDIS circumvents the guidance attenuation failure observed in joint-embedding methods when training data is scarce. Specifically, when data is limited to just 1%, DDIS maintains accuracy with a 40% advantage in error compared to joint approaches. These results demonstrate a significant advancement in reconstructing solutions from limited observations.

Measurements confirm that DDIS outperforms FunDPS, another method, in reconstructing the inverse Poisson problem under 1% paired data scarcity, exhibiting sharper and denser guidance. The l2 error for FunDPS was recorded at 54.12%, while DDIS achieved an error of 23.34%. Further tests on the forward problem showed DDIS achieving an l2 error of 3.80% and 1.08% respectively, indicating a high degree of accuracy in solution reconstruction.

The DDIS framework consists of three key components: diffusion prior learning in coefficient space, neural operator learning of the forward physics, and DAPS-based physics-aware posterior sampling. This decoupling allows for independent optimization of each component, leading to improved performance and data efficiency. The team’s work paves the way for more accurate and efficient solutions to inverse PDE problems, with potential applications in areas such as geophysics where forward computations are computationally expensive.

Decoupled modelling overcomes guidance attenuation in sparse inverse problem solving by learning a more robust solution

Scientists have developed a decoupled diffusion inverse solver (DDIS), a physics-aware generative framework designed for solving inverse partial differential equations (PDEs) with limited data. Unlike existing methods that jointly embed coefficient and solution information, DDIS separates the modelling of prior coefficients from physics-based likelihood evaluation using a neural operator.

This decoupling allows for more effective physics-informed learning and improved data efficiency in inverse problems. The key finding is that joint-embedding models struggle to provide reliable guidance when data is scarce or sensor layouts are sparse, experiencing guidance attenuation. DDIS addresses this by providing stable guidance through its neural operator, achieving state-of-the-art performance with improvements of 11% in error and 54% in spectral error under sparse observation.

Furthermore, the framework demonstrates robustness to resolution mismatch and maintains accuracy even when trained on low- or mixed-resolution data. The authors acknowledge that the performance of DDIS, while superior, is still dependent on the quality of the neural operator used for forward PDE solving.

They also note limitations related to the geometric conditions under which joint models fail, which require further investigation. Future research could explore the application of DDIS to more complex PDE systems and investigate methods for further enhancing the resolution-invariance of the neural operator, potentially broadening its applicability to a wider range of scientific computing challenges.