Physics Informed Differentiable Solvers for Learning Parametric Solution Manifolds in Heterogeneous Physical Systems
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912842804887552 |
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| author | Panahi, Milad Porta, Giovanni Michele Riva, Monica Guadagnini, Alberto |
| author_facet | Panahi, Milad Porta, Giovanni Michele Riva, Monica Guadagnini, Alberto |
| contents | Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application-oriented implications in fields such as hydrogeology where system properties exhibit significant (and often uncertain) spatial heterogeneity. We address this by reformulating a Physics-Informed Neural Network (PINN) as a differentiable solver that learns the continuous solution manifold for steady-state Darcy flow. Our framework requires only a single training run, circumventing the need for costly re-training for each new parameter instance. Its versatility is demonstrated through two representations of spatially heterogeneous hydraulic conductivity fields: a direct analytical form and a novel data-driven formulation resting on an autoencoder to create a low-dimensional latent encoding. A key innovation is the integration of the differentiable decoder into the physics-informed loss function, enabling on-the-fly reconstruction of complex conductivity fields via automatic differentiation. The approach yields accurate, mass-conserving flow solutions and supports efficient uncertainty quantification, providing a general methodology for physics-constrained data-driven modeling of heterogeneous systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_16350 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Physics Informed Differentiable Solvers for Learning Parametric Solution Manifolds in Heterogeneous Physical Systems Panahi, Milad Porta, Giovanni Michele Riva, Monica Guadagnini, Alberto Computational Physics 68T07 J.2; I.2.6; G.1.8; G.1.6 Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application-oriented implications in fields such as hydrogeology where system properties exhibit significant (and often uncertain) spatial heterogeneity. We address this by reformulating a Physics-Informed Neural Network (PINN) as a differentiable solver that learns the continuous solution manifold for steady-state Darcy flow. Our framework requires only a single training run, circumventing the need for costly re-training for each new parameter instance. Its versatility is demonstrated through two representations of spatially heterogeneous hydraulic conductivity fields: a direct analytical form and a novel data-driven formulation resting on an autoencoder to create a low-dimensional latent encoding. A key innovation is the integration of the differentiable decoder into the physics-informed loss function, enabling on-the-fly reconstruction of complex conductivity fields via automatic differentiation. The approach yields accurate, mass-conserving flow solutions and supports efficient uncertainty quantification, providing a general methodology for physics-constrained data-driven modeling of heterogeneous systems. |
| title | Physics Informed Differentiable Solvers for Learning Parametric Solution Manifolds in Heterogeneous Physical Systems |
| topic | Computational Physics 68T07 J.2; I.2.6; G.1.8; G.1.6 |
| url | https://arxiv.org/abs/2601.16350 |