Training with Hard Constraints: Learning Neural Certificates and Controllers for SDEs
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914363006255104 |
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| author | Kong, Chun-Wei Escobar, Sebastian Gracia, Ibon McMahon, Jay Lahijanian, Morteza |
| author_facet | Kong, Chun-Wei Escobar, Sebastian Gracia, Ibon McMahon, Jay Lahijanian, Morteza |
| contents | Due to their expressive power, neural networks (NNs) are promising templates for functional optimization problems, particularly for reach-avoid certificate generation for systems governed by stochastic differential equations (SDEs). However, ensuring hard-constraint satisfaction remains a major challenge. In this work, we propose two constraint-driven training frameworks with guarantees for supermartingale-based neural certificate construction and controller synthesis for SDEs. The first approach enforces certificate inequalities via domain discretization and a bound-based loss, guaranteeing global validity once the loss reaches zero. We show that this method also enables joint NN controller-certificate synthesis with hard guarantees. For high-dimensional systems where discretization becomes prohibitive, we introduce a partition-free, scenario-based training method that provides arbitrarily tight PAC guarantees for certificate constraint satisfaction. Benchmarks demonstrate scalability of the bound-based method up to 5D, outperforming the state of the art, and scalability of the scenario-based approach to at least 10D with high-confidence guarantees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23526 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Training with Hard Constraints: Learning Neural Certificates and Controllers for SDEs Kong, Chun-Wei Escobar, Sebastian Gracia, Ibon McMahon, Jay Lahijanian, Morteza Systems and Control Due to their expressive power, neural networks (NNs) are promising templates for functional optimization problems, particularly for reach-avoid certificate generation for systems governed by stochastic differential equations (SDEs). However, ensuring hard-constraint satisfaction remains a major challenge. In this work, we propose two constraint-driven training frameworks with guarantees for supermartingale-based neural certificate construction and controller synthesis for SDEs. The first approach enforces certificate inequalities via domain discretization and a bound-based loss, guaranteeing global validity once the loss reaches zero. We show that this method also enables joint NN controller-certificate synthesis with hard guarantees. For high-dimensional systems where discretization becomes prohibitive, we introduce a partition-free, scenario-based training method that provides arbitrarily tight PAC guarantees for certificate constraint satisfaction. Benchmarks demonstrate scalability of the bound-based method up to 5D, outperforming the state of the art, and scalability of the scenario-based approach to at least 10D with high-confidence guarantees. |
| title | Training with Hard Constraints: Learning Neural Certificates and Controllers for SDEs |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2602.23526 |