Learning to Solve Physics with Adaptive Coordinates

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Author: Denis Avetisyan

A new approach combines deep learning with spectral methods to find analytic solutions to physical problems with unprecedented efficiency.

The study demonstrates that a Vekua Cascade reconstruction consistently recovers smooth, analytic structures across diverse physics tasks-including those with sparse data and high frequencies where methods like Grid exhibit pixelation and SIREN struggles with convergence-highlighting its robustness and accuracy in complex signal recovery.
The study demonstrates that a Vekua Cascade reconstruction consistently recovers smooth, analytic structures across diverse physics tasks-including those with sparse data and high frequencies where methods like Grid exhibit pixelation and SIREN struggles with convergence-highlighting its robustness and accuracy in complex signal recovery.

This work introduces the Adaptive Vekua Cascade, a hybrid architecture leveraging coordinate-based neural networks and generalized analytic functions for a differentiable, physics-informed solver.

Coordinate-based neural networks, while promising for representing continuous physical fields, struggle with spectral bias and scalability to high-dimensional problems. This limitation motivates the development of ‘The Adaptive Vekua Cascade: A Differentiable Spectral-Analytic Solver for Physics-Informed Representation’, which introduces a hybrid architecture decoupling manifold learning from function approximation via a deep network that warps the physical domain onto a latent space suitable for analytic solutions. Crucially, the method employs a differentiable linear solver to optimally resolve spectral coefficients, achieving state-of-the-art accuracy with significantly reduced parameter counts. Does this approach represent a fundamental shift towards more memory-efficient and spectrally accurate scientific machine learning paradigms?

The Inherent Limitations of Dimensional Scaling

The computational demands of solving partial differential equations, such as the Navier-Stokes Equations governing fluid dynamics and the Helmholtz Equation describing wave phenomena, escalate dramatically with increasing dimensionality. This isn’t merely a linear increase; the complexity grows exponentially, a challenge often described as the “curse of dimensionality.” Simulating these equations requires discretizing the problem domain – essentially dividing it into a vast number of smaller elements – and solving for variables at each point. As the number of dimensions increases, the number of these elements explodes, demanding ever-greater computational resources – processing power, memory, and storage. For example, a simulation requiring $10^6$ grid points in two dimensions might necessitate $10^9$ points in three dimensions, and an impractical $10^{12}$ or more in four. Consequently, accurate modeling of many real-world phenomena, particularly those existing in higher-dimensional spaces, becomes prohibitively expensive or even impossible using traditional numerical methods.

The efficacy of many numerical methods diminishes rapidly as the dimensionality of a problem increases – a phenomenon known as the Curse of Dimensionality. This arises because the computational cost and data requirements grow exponentially with each added dimension. For instance, simulating a physical system governed by partial differential equations, such as fluid dynamics or electromagnetism, becomes prohibitively expensive as the number of spatial dimensions increases. Discretizing the solution domain – a standard technique in these methods – requires a number of computational points that scales exponentially with dimension, quickly exceeding available resources. Consequently, accurately modeling complex physical phenomena – like turbulent flow, high-dimensional materials, or multi-physics interactions – presents a significant challenge for traditional approaches, necessitating the development of alternative techniques capable of circumventing this fundamental limitation.

The challenges posed by high-dimensional partial differential equations are often compounded by a phenomenon known as spectral bias within many neural network solutions. These networks, while capable of approximating functions, tend to prioritize learning low-frequency components of the solution first – essentially, the “smooth” parts. This inherent preference can lead to inaccurate representations of high-frequency details crucial for resolving complex physical phenomena, especially when dealing with problems like turbulence or wave propagation. Consequently, even with sufficient training data, the network might converge to a solution that captures the overall trend but fails to accurately represent sharp gradients, oscillations, or fine-scale structures. This spectral bias isn’t necessarily a flaw, but rather a characteristic of common architectures, and understanding its impact is vital for developing more robust and reliable physics-informed neural networks capable of tackling challenging scientific computing problems, potentially requiring specialized architectures or training strategies to mitigate its effects and achieve solutions that accurately reflect the underlying physics described by equations like the $Helmholtz$ or $Navier-Stokes$ equations.

The Adaptive Vekua Cascade: A Framework for Dimensional Reduction

The Adaptive Vekua Cascade (AVC) is a computational framework designed for solving Partial Differential Equations (PDEs) that combines elements of both traditional numerical methods and deep learning. It utilizes Generalized Analytic Functions (GAFs), a class of functions with inherent properties suitable for representing solutions to a broad range of PDEs, including those with singularities or complex boundary conditions. Unlike conventional discretization techniques which approximate solutions at discrete points, the AVC employs GAFs to represent solutions in a continuous and differentiable manner. This allows for efficient computation of derivatives and integrals, critical for many PDE solution processes. The hybrid nature of the AVC aims to overcome limitations associated with purely data-driven approaches, by incorporating prior knowledge about the underlying physics encoded within the GAF representation, while still leveraging the learning capabilities of neural networks to adapt to specific problem instances.

Coordinate-Based Neural Networks (CBNNs) within the Adaptive Vekua Cascade address the challenges of directly discretizing partial differential equations (PDEs) by learning a non-linear mapping from physical coordinates to a lower-dimensional latent space. Traditional discretization methods, such as finite element or finite difference schemes, require a fixed grid and can struggle with complex geometries or high-dimensional problems. CBNNs, however, operate directly on the coordinate space, allowing the network to learn a representation where the solution to the PDE is smoother and more efficiently represented. This is achieved by training the CBNN to map input coordinates $x \in \mathbb{R}^n$ to latent vectors $z \in \mathbb{R}^m$, where $m < n$, effectively reducing the dimensionality and computational cost associated with solving the PDE. The network learns this mapping through supervised training, minimizing the error between the predicted solution in the latent space and the true solution, thereby circumventing the limitations of fixed-grid discretization.

Deep Coordinate Warping (DCW) within the Adaptive Vekua Cascade (AVC) operates by learning a deformation field that maps points in a regular computational domain to the physical domain of the problem. This learned mapping, represented by a neural network, allows the AVC to concentrate computational effort in regions of high solution gradient or complex geometry, improving accuracy and efficiency. DCW effectively decouples the discretization of the physical domain from the solution representation, enabling the use of a fixed, regular grid for network evaluation while still accurately capturing complex solution features. The warping field is learned concurrently with the solution mapping, minimizing reconstruction error and ensuring a consistent transformation between coordinate spaces. This process results in a more efficient representation of the solution compared to direct discretization methods, particularly for problems with singularities or highly non-uniform features.

Efficient Linear Solvers and Optimization Strategies

The AVC employs a Differentiable Linear Solver to determine the spectral coefficients representing data within the latent space. This allows for the computation of gradients with respect to these coefficients, facilitating optimization via gradient descent. Unlike traditional linear solvers which are non-differentiable and require iterative methods, the AVC’s implementation enables direct backpropagation through the linear solving process. This is crucial for end-to-end training, as it permits the adjustment of all model parameters – including those defining the linear system – based on the loss function. The solver computes $x = A^{-1}b$ where $x$ represents the optimal spectral coefficients, $A$ is the linear operator, and $b$ is the target data, all within the gradient computation graph.

The AVC’s linear solver utilizes Ridge Regression, a least squares regression modified by the addition of a regularization parameter, to address potential multicollinearity and improve the conditioning of the system of equations. This regularization minimizes the impact of noisy or highly correlated input features. To solve the resulting normal equations efficiently and maintain numerical stability, Cholesky Decomposition is employed. This decomposition factorizes the symmetric, positive-definite covariance matrix into the product of a lower triangular matrix and its transpose, reducing computational complexity and avoiding the need for computationally expensive matrix inversions. The combined approach of Ridge Regression and Cholesky Decomposition ensures both the accuracy and stability of the spectral coefficient computation, particularly crucial for gradient-based optimization within the AVC framework.

Optimization of the AVC process is achieved through the utilization of an Optax optimizer, a framework designed for scalable and flexible gradient-based optimization. Optax facilitates end-to-end training by automating the application of gradient updates to all trainable parameters within the linear solver and spectral coefficient computation stages. This allows for joint refinement of the entire solution, rather than requiring separate optimization loops for individual components. Supported optimization algorithms within Optax include Adam, SGD, and Adafactor, enabling selection of an appropriate method based on specific performance requirements and computational constraints. The framework also supports features such as learning rate scheduling, momentum, and weight decay, further enhancing the optimization process and facilitating convergence to an optimal solution.

Validation and the Circumvention of Practical Limitations

The Analytic Variance Component (AVC) method showcases a remarkable ability to reconstruct images even when data is severely limited, a capability rigorously tested using the Shepp-Logan phantom. This phantom, a standard benchmark in image reconstruction, presents a challenging scenario with discrete features and varying densities, allowing for precise evaluation of an algorithm’s performance. Results indicate the AVC consistently produces high-fidelity reconstructions from significantly fewer data points than conventional methods, effectively addressing the ill-posed nature of the reconstruction problem. This success stems from the AVC’s innovative approach to variance modeling, which intelligently prioritizes the most informative data and mitigates the impact of missing or noisy samples, resulting in clearer and more accurate images even under challenging conditions.

Conventional approaches to reconstructing complex data often stumble upon the Curse of Dimensionality, where the volume of possible data grows exponentially with the number of dimensions, rendering computations intractable. The Analytic Variational Computation (AVC) circumvents this issue through its intrinsic design, unlike methods dependent on Multi-Resolution Hash Encodings. These encoding techniques, while attempting to manage complexity, can still falter as dimensionality increases. The AVC, however, achieves efficient representation by leveraging analytic basis expansions with harmonic functions, effectively compressing information without the exponential growth in computational cost. This allows the AVC to scale effectively to high-dimensional problems, maintaining accuracy and efficiency even with limited data, and ultimately offering a pathway to solve problems previously inaccessible due to computational limitations.

The Analytic Variational Computation (AVC) method achieves high-fidelity solutions by directly tackling the issue of spectral bias – a common limitation in neural network-based approaches where low-frequency components are preferentially learned. Techniques such as Fourier Feature Mappings are employed to enhance representation, but the AVC intrinsically mitigates this bias through its use of analytic basis expansion with harmonic basis functions. This approach not only improves accuracy but also dramatically reduces computational cost; as demonstrated in Experiment E, simulating Navier-Stokes equations with the AVC required only 840 parameters to achieve state-of-the-art results, a figure dwarfed by the 4.2 million parameters necessary for traditional grid-based methods. This represents a substantial parameter reduction, effectively circumventing the curse of dimensionality and enabling efficient simulations of complex spatiotemporal physics.

A core achievement of this research lies in its dramatic reduction of necessary parameters for modeling complex spatiotemporal physics. Traditional grid-based methods often require an exponential increase in parameters to accurately represent increasingly detailed phenomena, a limitation known as the Curse of Dimensionality. However, the Analytic Variational Computation (AVC) framework demonstrably circumvents this issue; in experiments simulating Navier-Stokes equations, AVC achieved state-of-the-art accuracy with merely 840 parameters, a figure dwarfed by the 4.2 million required by conventional grid-based approaches. This substantial parameter reduction isn’t simply a computational efficiency; it signifies a fundamental shift, allowing for the modeling of highly complex physical systems with significantly less data and computational resources, effectively breaking the barriers previously imposed by the Curse of Dimensionality in this domain.

The Adaptive Vekua Cascade, as presented, pursues a fundamentally elegant solution to the challenges of scientific machine learning. It inherently acknowledges that the representation of a problem – the coordinate system, in this case – dramatically impacts solvability. This resonates deeply with a principle articulated by Marvin Minsky: “You can’t always get what you want; but if you try sometimes you find, you get what you need.” The AVC doesn’t simply attempt to solve physical problems; it actively learns the coordinate transformation needed to reveal the analytic solution, effectively shaping the problem itself to facilitate accurate and efficient computation. The architecture’s focus on spectral bias and generalized analytic functions demonstrates a commitment to mathematical purity, striving for solutions that are provable through their inherent structure, rather than merely empirically successful.

What’s Next?

The Adaptive Vekua Cascade, while demonstrating a commendable alignment of learned representations with established analytic solutions, merely scratches the surface of a deeper challenge. The architecture’s reliance on coordinate system learning, though effective, begs the question: are optimal coordinate systems merely a convenient trick, or do they reflect a fundamental property of the underlying physics? If the latter, then the pursuit shifts from learning a coordinate system to discovering the coordinate system – a task demanding a more rigorous theoretical framework than current gradient-based methods provide. If it feels like magic that a neural network can stumble upon something resembling analytic continuation, it is because the invariant has not yet been fully revealed.

Further work must address the limitations inherent in spectral methods themselves. The curse of dimensionality remains a persistent threat, and the AVC’s performance will inevitably degrade as problem complexity increases. Exploring hybrid architectures that combine the strengths of spectral, finite element, and meshfree methods, guided by learned error estimators, represents a promising avenue for future research. The true test lies not in achieving high accuracy on benchmark problems, but in applying this framework to genuinely intractable systems where analytical solutions are, and will remain, out of reach.

Ultimately, the field must move beyond simply solving equations to understanding the solutions. The pursuit of differentiable physics is not merely an exercise in numerical efficiency; it is a quest for a more profound, and mathematically elegant, description of the natural world. The AVC offers a glimpse of this future, but much work remains to transform this glimpse into a clear and unwavering vision.

Original article: https://arxiv.org/pdf/2512.11776.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-16 00:32